} 


'^f...^^t2^''p^-t!~^^ 


ELEMENT AET 


Mathematical  Tables. 


BY 

ALEXANDER   MACFAELANE,   D.Sc,  LL.D., 

Pbopessoe  of  Physics  in  the  Univebsitt  op  Texas. 


Boston,  U.S.A.,  and  London: 

GINN  &  COMPANY,   PUBLISHERS. 

1894. 


Entired  at  Stationers'  Hall. 


Entered,  according  to' Act  of  Congress,  in  the  year  1889,  bv 

ALEXANDER  MACFARLANE, 
in  the  OflBce  of  the  Librarian  of  Congress,  at  Washington, 


All  Rights  Reserved. 

ASTBoiKanr  DIPT. 


Ttpographt  by  J.  8.  Gushing  &  Co.,  Boston,  U.S.A. 
Presswobk  by  Ginn  &  Co.,  Boston,  U.S.A. 


;  /\  _L 


PREFACE. 


rr^HESE  tables  are  designed  to  be  useful  not  only  in  com- 
-*-  puting  and  in  the  graphic  method,  but  also  in  the 
teaching  of  arithmetic  and  in  the  illustration  of  the  theo- 
rems of  algebra. 

I  have  arranged  the  several  tables  on  a  uniform  decimal 
plan,  so  that  the  entries  for  a  particular  number  are  generally 
found  in  the  same  position  on  the  page.  The  arrangement 
is  that  of  double  entry,  so  that  in  general  the  order  of 
reading  is  the  same  as  for  ordinary  print.  The  argument 
and  entry  are  so  expressed  that  it  is  easy  to  find  the  entry 
corresponding  to  any  other  position  of  the  decimal  point  in. 
the  argument.  In  most  cases  the  whole  of  a  table  is  seen 
at  one  opening  of  the-  pages,  and  the  tenth  compartment, 
when  not  required  for  the  main  table,  is  filled  with  a  short 
table  which  is  in  general  auxiliary  to  the  main  table. 

Special  acknowledgments  are  due  to  Prof.  Hastings  of 
Yale  University,  and  Prof.  Halsted  of  this  University.  In 
the  proof-reading  and  independent  computation,  I  have  re- 
ceived aid  from  D.  W.  Spence  and  J.  C.  Nagle,  science 
students  of  this  University. 

Alexander  Macfaklaxe. 
University  of  Texas, 
April,  1889. 


M298746 


CONTENTS. 


PAGES 

I.     Common  logarithms.     Proportional  parts,*  4-23       .  2-3 

The  same  for  1000-1900.     Logarithms  for  interest     .  4-5 

II.     Aiitilogarithms 6-7 

III.  Addition  logarithms 8-9 

IV.  Subtraction  logarithms.     Proportional  parts,  1-38  .  10-13 
Y.     Logarithmic  sines  and  cosines.     Proportional  parts, 

11-30 14-1.5 

The  same  for  0°.00  to  9°. 00.     Equivalent  of  degrees 

in  minutes.      Table  of  S  for  degrees     .         .         .     16-17 
YI.     Logarithmic  tangents  and  cotangents.    Proportional 

parts,  31-50 18-19 

The  same  for  0°.00  to  9°.00.     Equivalent  of  minutes 

in  degrees.     Table  of  T  for  degrees    .         .         .     20-21 
YIL     Logarithmic  sines  and  cosines  for  minutes.    Propor- 
tional parts,  l~^:Q.     Table  of  S  for  minutes          .     22-23 
VIII.     Logarithmic  tangents  and  cotangents  for  minutes. 
Proportional  parts,  50-99.      Table  of  T  for  min- 
utes         24-25 

IX.     Natural  sines  and  cosines.    Proportional  pai-ts  to  six, 

1-19 26-27 

X.     Natural   tangents   and    cotangents.      Proportional 

parts  to  six,  20-59 28-29 

XL     Natural  secants  and  cosecants.     Proportional  parts 

to  six,  60-99 30-31 

XII.  Radians  equivalent  to  degrees.  Radians  equivalent 
to  m,inutes.  Fraction  of  period,  and  time,  equiva- 
lent to  degrees 32-33 


*  A  caption  in  Italics  indicates  a  small  table  appended  to  the  main 
table. 


CONTENTS. 


PAGES 

Xni.     Reciprocals,     Multiples  of  ^  to  ^^     .        .        ,        .  34-35 

XIV.     Squares.     Square  of  reciprocal 36-37 

XV.     Cubes.     Cube  of  reciprocal 38-39 

XVI.     Square  roots.     Square  root  of  reciprocal   .        .         .  40-43 

XVII.     Cube  roots.     Cube  root  of  reciprocal          .         .         .  44-49 

XVIII.     Multiples  of  100-999 50-67 

XIX.     Circumference  of  circle.     Constants  involving  ir        .  68-69 

XX.     Area  of  circle.     Multiples  of -^^  and  ^^     .         .         .  70-71 
XXI.     Content  of  sphere.     Logarithms  of  factorials  and  of 

Powersoft 72-73 

XXII.     Hyperbolic  logarithms.     Multiples  of  the  modulus    .  74-75 

XXIII.  Amount  at  the  end  of  n  years           ....  76 

XXIV.  Present  value  of  1000 77 

XXV.     Amount  of  an  annuity  paid  at  the  end  of  each  year  78 

XXVI.     Present  value  of  the  preceding         ....  79 
XXVII.     Amount  of  an  annuity  paid  at  the  beginning  of 

each  year 80 

XXVIII.     Sum  to  be  paid  at  the  end  of  each  of  n  years  to 

extinguish  a  debt  of  1000          ....  81 

XXIX.     Least  divisors 82-85 

XXX.     Exponentials 86 

Explanation  of  Tables 87-105 

XXXI.    Multiples  of  1-100 Folding  leaf 


I.   COMMON  LOGAEITHMS. 


logn 


n 

0 

1   2   3 

4   5   0 

7   8   9 

d 

10 

0000 

0043  0086  0128 

0170  0212  0253 

0294  0334  0374 

40 

11 
12 
13 

0414 
0792 
1139 

0453  0492  0531 
0S2S  0864  0899 
1173  1206  1239 

0569  0607  0645 
0934  0969  1004 
1271  1303  1335 

0682  0719  0755 
1038  1072  1106 
1367  1399  1430 

37 
33 
31 

U 
15 
IG 

1461 
1761 
2041 

1492  1523  1553 
1790  1818  1847 
2068  2095  2122 

1584  1614  1644 
1875  1903  1931 
2148  2175  2201 

1673  1703  1732 
1959  1987  2014 
2227  2253  2279 

29 
27 

25 

17 

IS 
19 

2304 
2553 
27SS 

2330  2355  2380 
2577  260i  2625 
2810  2833  2856 

2405  2430  2455 
2648  2672  2695 
2878  2900  2923 

2480  2504  2529 
2718  2742  2765 
2945  2967  2989 

24 
23 
21 

20 

3010 

3032  3054  3075 

3096  3118  3139 

3160  3181  3201 

21 

21 
22 
23 

3222 
3424 
3617 

3243  3263  3284 
3444  3464  3483 
3636  3655  3674 

3304  3324  3345 
3502  3522  3541 
3692  3711  3729 

3365  3385  3404 
3560  3579  3598 
3747  3766  3784 

20 
19 
18 

24 
25 
2G 

3802 
3979 
4150 

3820  3838  3856 
3997  4014  4031 
4166  4183  4200 

3874  3892  3909 
4048  4065  4082 
4216  4232  4249 

3927  3945  3962 
4099  4116  4133 
4265  4281  4298 

17 
17 
16 

27 

28 
29 

4314 
4472 
4624 

4330  4346  4362 
4487  4502  4518 
4639  4654  4669 

4378  4393  4409 
4533  4548  4564 
4683  4698  4713 

4425  4440  4456 
4579  4594  4609 
4728  4742  4757 

16 

15 
14 

30 

4771 

4786  4800  4814 

4829  4843  4857 

4871  4886  4900 

14 

31 
32 
33 

4914 
5051 
5185 

4928  4942  4955 
5065  5079  5092 
5198  5211  5224 

4969  4983  4997 
5105  5119  5132 
5237  5250  5263 

5011  5024  5038 
5145  5159  5172 
5276  5289  5302 

13 
13 
13 

34 
35 
36 

5315 
5441 
5563 

5328  5340  5353 
5453  5465  5478 
5575  5587  5599 

5366  5378  5391 
5490  5502  5514 
5611  5623  5635 

5403  5416  5428 
5527  5539  5551 
5647  5658  5670 

13 
12 
12 

37 
38 
39 

5682 
5798 
5911 

5694  5705  5717 
5809  5821  5832 
5922  5933  5944 

5729  5740  5752 
5843  5855  5866 
5955  5966  5977 

5763  5775  5786 
5877  5888  5899 
5988  5999  6010 

12 
12 
11 

40 

6021 

6031  6042  6053 

6064  6075  6085 

6096  6107  6117 

11 

41 
42 
43 

6128 
6232 
633i 

6138  6149  6160 
6243  6253  6263 
6345  6355  6365 

6170  6180  6191 
6274  6284  6294 
6375  6385  6395 

6201  6212  6222 
6304  6314  6325 
6405  6415  6425 

10 
10 
10 

44 
45 
46 

6435 
6532 
6628 

6444  6454  6464 
6542  6551  6561 
6637  6646  6656 

6474  6484  6493 
6571  6580  6590 
6665  6675  6684 

6503  6513  6522 
6599  6609  6618 
6693  6702  6712 

10 
10 
9 

47 

48 
49 

6721 
6S12 
6902 

6730  6739  6749 
6821  6830  6839 
691]  6920  6928 

6758  6767  6776 
6848  6857  6866 
6937  6946  6955 

6785  6794  6803 
6875  6884  6893 
6964  6972  6981 

9 
9 
9 

50 

6990 

6998  7007  7016 

7024  7033  7042 

7050  7059  7067 

9 

61 
52 
53 

7076 
7160 
7243 

7084  7093  7101 
7168  7177  7185 
7251  7259  7267 

7110  7118  7126 
7193  7202  7210 
7275  7284  7292 

7135  7143  7152 
7218  7226  7235 
7300  7308  7316 

8 
8 
8 

54 
55 
56 

7324 
7404 
7482 

7332  7340  7348 
7412  7419  7427 
7490  7497  7505 

7356  7364  7372 
7435  7443  7451 
7513  7520  7528 

7380  7388  7396 
7459  7466  7474 
7536  7543  7551 

8 
8 
8 

57 

58 
69 

7559 
7634 
77,09 

7566  7574  7582 
7642  7649  7657 
7716  7723  7731 

7589  7597  7604 
7664  7672  7679 
7738  7745  7752 

7612  7619  7627 
7686  7694  7701 
7760  7767  7774 

7 
8 
8 

lOgfl 


I.   COMMON   LOQAEITHMS. 


n 

0 

12   3 

4   5   6 

7   8   9 

d 

60 

7782 

7789  7796  7803 

7810  7818  7825 

7832  7839  7846 

7 

61 

7853 

7860  7868  7875 

7882  7889  7896 

7903  7910  7917 

7 

62 

7924 

7931  7938  7945 

7952  7959  7966 

7973  7980  7987 

6 

63 

7993 

8000  8007  8014 

8021  8028  8035 

8041  8048  8055 

7 

64 

8062 

8069  8075  8082 

8089  8096  8102 

8109  8116  8122 

7 

65 

8129 

8136  8142  8149 

8156  8162  8169 

8176  8182  8189 

'  6 

66 

8195 

8202  8209  8215 

8222  8228  8235 

8241  8248  8254 

7 

67 

8261 

8267  8274  8280 

8287  8293  8299 

8306  8312  8319 

6 

68 

8325 

8331  8338  8344 

8351  8357  8363 

8370  8376  8382 

6 

69 

8388 

8395  8401  8407 

8414  8420  8426 

8432  8439  8445 

6 

70 

8451 

8457  8463  8470 

8476  8482  8488 

8494  8500  8506 

7 

71 

8513 

8519  8525  8531 

8537  8543  8549 

8555  8561  8567 

6 

72 

8573 

8579  8585  8591 

8597  8603  8609 

8615  8621  8627 

6 

73 

8633 

8639  8645  8651 

8657  8663  8669 

8675  8681  8686 

6 

74 

8692 

S698  8704  8710 

8716  8722  8727 

8733  8739  8745 

6 

75 

8751 

8756  8762  8768 

8774  8779  8785 

8791  8797  8802 

6 

76 

8808 

8814  8820  8825 

SS31  8S37  8842 

8848  8854  8859 

6 

77 

8865 

8871  8876  8882 

8SS7  8893  8899 

8904  8910  8915 

6 

78 

8921 

8927  8932  8938 

8943  8949  8954 

8960  8965  8971 

5 

79 

8976 

8982  8987  8993 

8998  9004  9009 

9015  9020  9025 

6 

80 

9031 

9036  9042  9047 

9053  9058  9063 

9069  9074  9079 

6 

81 

9085 

9090  9096  9101 

9106  9112  9117 

9122  9128  9133 

82 

9138 

9143  9149  9154 

9159  9165  9170 

9175  9180  9186 

83 

9191 

9196  9201  9206 

9212  9217  9222 

9227  9232  9238 

84 

9243 

9248  9253  9258 

9263  9269  9274 

9279  9284  9289 

85 

9294 

9299  9304  9309 

9315  9320  9325 

9330  9335  9340 

86 

9345 

9350  9355  9360 

9365  9370  9375 

9380  9385  9390 

87 

9395 

9400  9405  9410 

9415  9420  9425 

9430  9435  9440 

88 

9445 

9450  9455  9460 

9465  9469  9474 

9479  9484  9489 

89 

9494 

9499  9504  9509 

9513  9518  9523 

9528  9533  9538 

90 

9542 

9547  9552  9557 

9562  9566  9571 

9576  9581  9586 

91 

9590 

9595  9600  9605 

9609  9614  9619 

9624  9628  9633 

92 

9638 

9643  9647  9652 

9657  9661  9666 

9671  9675  9680 

93 

9685 

9689  9694  9699 

9703  9708  9713 

9717  9722  9727 

94 

9731 

9736  9741  9745 

9750  9754  9759 

9763  9768  9773 

95 

9777 

9782  9786  9791 

9795  9800  9805 

9809  9814  9818 

96 

9823 

9827  9832  9836 

9841  9845  9850 

9854  9859  9S63 

97 

9868 

9872  9877  9881 

9886  9890  9894 

9899  9903  9908 

98 

9912 

9917  9921  9926 

9930  9934  9939 

9943  9948  9952 

99 

9956 

9961  9965  9969 

9974  9978  9983 

9987  9991  9996 

pp. 

4  5  6789 

10 

11  12 

13  ,  14  15  16 

17  18  19 

20 

21  22  23 

1 

0  1  1 

1  1  1 

T 

1  1 

1  2  2 

~2 

2  2 

~2 

2  2  2 

2 

1  1  1 

1  2  2 

2 

2  2 

3     3     3 

3 

4  4 

4 

4  4  5 

3 

1  2  2 

2  2  3 

3 

3  4 

4  5  5 

5 

5  6 

6 

6  7  7 

4 

2  2  2 

3  3  4 

4 

4  5 

6  6  6 

1 

7  8 

8 

8  9  9 

5 

2  3  3 

4  4  5 

S 

6  6 

7  8  8 

9 

9  10 

10 

11  11  12 

6 

2  3  4 

4  5  5 

6 

7  7 

8 

8  9  10 

IC 

11  11 

12 

13  13  14 

7 

3  4  4 

5  6  6 

7 

8  8 

9 

10  11  11 

12 

13  13 

14 

15  15  16 

8 

3  4  5 

6  6  7 

8 

9  10 

10 

11  12  13 

14 

14  15 

16 

17  18  18 

9 

4  5  5 

6  7  8 

9 

10  11 

12  13  14  14 

IS 

16  17il8 

19  20  21 

I.   COMMON  LOGAEITHMS 


logn 


n 

0 

12   3 

4   5   6 

7    8   9 

d 

pp. 

100 

0000 

0004  0009  0013 

0017  0022  0026 

0030  0035  0039 

4 

5 

101 

102 
103 

0043 
00S6 
012S 

0048  0052  0056 
0090  0095  0099 
0133  0137  0141 

0060  0065  0069 
0103  0107  0111 
0145  0149  0154 

0073  0077  0082 
0116  0120  0124 
0158  0162  qi66 

4 
4 
4 

1 
1 
2 

104 
105 
100 

0170 
0212 
0253 

0175  0179  0183 
0216  0220  0224 
0257  0261  0265 

0187  0191  0195 
0228  0233  0237 
0269  0273  0278 

0199  0204  0208 
0241  0245  0249 
0282  0286  0290 

4 
4 
4 

2 
3 
3 

107 
lOS 
109 

0294 
0334 
0374 

029S  0302  0306 
0338  0342  0346 
0378  0382  0386 

0310  0314  0318 
0350  0354  0358 
0390  0394  0398 

0322  0326  0330 
0362  0366  0370 
0402  0406  0410 

4 
4 
4 

4 
4 
5 

110 

0414 

041S  0422  0426 

0430  0434  0438 

0441  0445  0449 

4 

111 
112 
113 

0453 
0492 
0531 

0457  0461  0465 
0496  0500  0504 
0535  0538  0542 

0469  0473  0477 
0508  0512  0515 
0546  0550  0554 

0481  0484  0488 
0519  0523  0527 
0558  0561  0565 

4 

4 
4 

lU 
115 

IIG 

0569 
0607 
0645 

0573  0577  05S0 
0611  0615  0618 
0648  0652  0656 

0584  0588  0592 
0622  0626  0630 
0660  0663  0667 

0596  0599  0603 
0633  0637  0641 
0671  0674  0678 

4 

4 
4 

117 
118 
119 

06S2 
0719 
0755 

0686  0689  0693 
0722  0726  0730 
0759  0763  0766 

0697  0700  0704 
0734  0737  0741 
0770  0774  0777 

0708  0711  0715 
0745  0748  0752 
0781  0785  0788 

4 
3 
4 

120 

0792 

0795  0799  0803 

0806  0810  0813 

0817  0821  0824 

4 

4 

121 
122 
123 

0S28 
0S64 
0899 

083 1  0835  0839 
0867  0S71  0874 
0903  0906  0910 

0842  0846  0849 
0878  0881  0885 
0913  0917  0920 

0853  0856  0860 
0888  0892  0896 
0924  0927  0931 

4 
3 
3 

0 

1 
1 

124 
125 
126 

0934 
0969 
1004 

0938  0941  0945 
0973  0976  0980 
1007  1011  1014 

0948  0952  0955 
0983  0986  0990 
1017  1021  1024 

0959  0962  0966 
0993  0997  1000 
1028  1031  1035 

3 
4 
3 

2 
2 
2 

127 
128 
129 

1038 
1072 
1106 

1041  1045  1048 
1075  1079  1082 
1109  1113  1116 

1052  1055  1059 
1086  1089  1092 
1119  1123  1126 

1062  1065  1069 
1096  1099  1103 
1129  1133  1136 

3 
3 
3 

3 
3 
4 

130 

1139 

1143  1146  1149 

1153  1156  1159 

1163  1166  1169 

4 

131 
132 
133 

1173 
1206 
1239 

1176  1179  1183 
1209  1212  1216 
1242  1245  1248 

1186  1189  1193 
1219  1222  1225 
1252  1255  1258 

1196  1199  1202 
1229  1232  1235 
1261  1265  1268 

4 
4 
3 

134 
135 
13G 

1271 
1303 
1335 

1274  1278  1281 
1307  1310  1313 
1339  1342  1345 

1284  1287  1290 
1316  1319  1323 
1348  1351  1355 

1294  1297  1300 
1326  1329  1332 
1358  1361  1364 

3 
3 
3 

137 
138 
139 

1367 
1399 
1430 

1370  1374  1377 
1402  1405  1408 
1433  1436  1440 

1380  1383  1386 
1411  1414  1418 
1443  1446  1449 

1389  1392  1396 
1421  1424  1427 
1452  1455  1458 

3 
3 
3 

140 

1461 

1464  1467  1471 

1474  1477  1480 

1483  1486  1489 

3 

3 

141 
142 
143 

1492 

1523 
1553 

1495  1498  1501 
1526  1529  1532 
1556  1559  1562 

1504  1508  1511 
1535  1538  1541 
1565  1569  1572 

1514  1517  1520 
1544  1547  1550 
1575  1578  1581 

3 
3 
3 

0 

1 
1 

144 
145 
140 

1584 
1614 
1644 

1587  1590  1593 
1617  1620  1623 
1647  1649  1652 

1596  1599  1602 
1626  1629  1632 
1655  1658  1661 

1605  1608  1611 
1635  1638  1641 
1664  1667  1670 

3 
3 
3 

1 
2 
2 

147 
148 
149 

1673 
1703 
1732 

1676  1679  1682 
1706  1708  1711 
1735  1738  1741 

1685  1688  1691 
1714  1717  1720 
1744  1746  1749 

1694  1697  1700 
1723  1726  1729 
1752  1755  1758 

3 
3 
3 

2 
2 
3 

logn 


CONTINUED. 


n 

0 

12    3 

4   5   6 

7    8   9 

d 

pp. 

150 

1761 

1764  1767  1770 

1772  1775  1778 

1781  1784  1787 

3 

3 

151 
152 
153 

1790 
1818 
1847 

1793  1796  1798 
1821  1824  1827 
1850  1853  1855 

ISOl  1804  1807 
1830  1833  1836 
1858  1861  1864 

1810  1813  1816 
1838  1841  1844 
1867  1870  1872 

2 
3 
3 

0 
1 

1 

154 
155 
156 

1875 
1903 
1931 

1878  1881  1SS4 
1906  1909  1912 
1934  1937  1940 

1886  1889  1892 
1915  1917  1920 
1942  1945  1948 

1895  1898  1901 
1923  1926  1928 
1951  1953  1956 

2 
3 
3 

1 
2 
2 

157 
158 
159 

1959 
1987 
2014 

1962  1965  1967 
1989  1992  1995  . 
2017  2019  2022 

1970  1973  1976 
1998  2000  2003 
2025  2028  2030 

1978  1981  1984 
2006  2009  2011 
2033  2036  2038 

3 
3 
3 

2 
2 
3 

160 

2041 

2044  2047  2049  !  2052  2055  2057 

2060  2063  2066 

2 

161 
162 
163 

2068 
2095 
2122 

2071  2074  2076 
2098  2101  2103 
2125  2127  2130 

2079  2082  2084 
2106  2109  2111 
2133  2135  2138 

2087  2090  2092 
2114  2117  2119 
2140  2143  2146 

3 
3 

2 

164 
165 
166 

2148 
2175 
2201 

2151  2154  2156 
2177  2180  2183 
2204  2206  2209 

2159  2162  2164 
2185  2188  2191 
2212  2214  2217 

2167  2170  2172 
2193  2196  2198 
2219  2222  2225 

3 
3 

2 

167 
168 
169 

2227 
2253 
2279 

2230  2232  2235 
2256  2258  2261 
2281  2284  22S7 

2238  2240  2243 
2263  2266  2269 
2289  2292  2294 

2245  2248  2251 
2271  2274  2276 
2297  2299  2302 

2 
3 
2 

170 

2304 

2307  2310  2312 

2315  2317  2320 

2322  2325  2327 

3 

2 

171 
172 
173 

2330 

2355 
2380 

2333  2335  2338 
2358  2360  2363 
2383  2385  2388 

2340  2343  2345 
2365  2368  2370 
2390  2393  2395 

2348  2350  2353 
2373  2375  2378 
2398  2400  2403 

2 
2 
2 

0 
0 
1 

174 
175 
176 

2405 
2430 
2455 

2408  2410  2413 
2433  2435  2438 
2458  2460  2463 

2415  2418  2420 
2440  2443  2445 
2465  2467  2470 

2423  2425  2428 
2448  2450  2453 
2472  2475  2477 

2 
2 
3 

1 
1 
1 

177 

178 
179 

180 

2480 
2504 
2529 

2482  2485  2487 
2507  2509  2512 
2531  2533  2536 

2490  2492  2494 
2514  2516  2519 
2538  2541  2543 

2497  2499  2502 
2521  2524  2526 
2545  2548  2550 

2 
3 
3 

1 
2 
2 

2553 

2555  2558  2560 

2562  2565  2567 

2570  2572  2574 

3 

181 

182 
183 

2577 
2601 
2625 

2579  2582  2584 
2603  2605  2608 
2627  2629  2632 

2586  2589  2591 
2610  2613  2615 
2634  2636  2639 

2594  2596  2598 
2617  2620  2622 
2641  2643  2646 

3 
3 

2 

184 
185 
186 

2648 
2672 
2695 

2651  2653  2655 
2674  2676  2679 
2697  2700  2702 

2658  2660  2662 
2681  2683  2686 
2704  2707  2709 

2665  2667  2669 
2688  2690  2693 
2711  2714  2716 

3 
2 

2 

187 
188 
189 

2718 
2742 

2765 

2721  2723  2725 
2744  2746  2749 
2767  2769  2772 

2728  2730  2732 
2751  2753  2755 
2774  2776  2778 

2735  2737  2739 
2758  2760  2762 
2781  2783  2785 

3 
3 
3 

n 

0 

1   2   3 

4   5   6 

7   8   9] 

d 

430 

1.00 

.00  000 

0 

0434  0S6S  1301 

1734  2166  2598 

3029  3461  38 

^1 

1.01 
1.02 
1.03 

.00  432 
.00  860 
.01  283 

1 
0 

7 

4751  5181  5609 
9026  9451  9S76 
3259  3680  4100 

6038  6466  6894 

*0300*0724*1147 

4521  4940  5360 

7321  7748  81 

*1570*1993*24 

5779  6197  66 

74 
15 
16 

426 
422 
417 

1.04 
1.05 
1.06 

.01  703 
.02  lis 
.02  53C 

3 
9 
6 

7451  7868  8284 
1603  2016  2428 
5715  6125  6533 

8700  9116  9532 
2841  3252  3664 
6942  7350  7757 

9947*0361*07 
4075  4486  48 
8164  8571  89 

75 
96 

78 

414 

410 
406 

1.07  |.02  935 
1.081.03  342 
1.091.03  742 

4 
4 
6 

9789*0195*0600 
3826  4227  4628 
7825  8223  8620 

*1004*140S*1812 
5029  5430  5830 
9017  9414  9811 

*22] 6*2619*30 
6230  6629  70 
*0207*0602*09 

21 
28 
98 

403 
398 
395 

11.   ANTILOGAEITHMS. 


10  " 


In 

0 

12   3 

4   5 

(J 

7 

8   9 

d 

pp. 

00 

1000 

1002  1005  1007 

1009  1012 

1014 

1016 

1019  1021 

2 

234 

01 
02 
03 

1023 
1047 
1072 

1026  1028  1030 
1050  1052  1054 
1074  1076  1079 

1033  1035 
1057  1059 
1081  1084 

1038 
1062 
1086 

10-10  1042  1045 
1064  1067  1069 
1089  1091  1094 

2 
3 

2 

000 
01  1 

1 1 1 

04 
05 
00 

1096 
1122 
IHS 

1099  1102  1104 
1125  1127  1130 
115l  1153  1156 

1107  1109 
1132  1135 
1159  1161 

1112 
1138 
1164 

1114 
1140 
1167 

1117  1119 
1143  1146 
1169  1172 

3 
2 
3 

1  1  2 
1  22 
122 

07 
08 
09 

1175 
1202 
1230 

1178  1180  11S3 
1205  1208  1211 
1233  1236  1239 

1186  1189 
1213  1216 
1242  1245 

1191 
1219 
1247 

1194 
1222 
1250 

1197  1199 
1225  1227 
1253  1256 

3 
3 
3 

123 
223 
234 

10 

1259 

1262  1265  1268 

1271  1274 

1276 

1279 

1282  1285 

3 

56  7 

11 
12 
13 

12SS 
131S 
1349 

1291  1294  1297 
1321  1324  1327 
1352  1355  1358 

1300  1303 
1330  1334 
•1361  1365 

1306 
1337 
1368 

1309 
1340 
1371 

1312  1315 
1343  1346 
1374  1377 

3 
3 
3 

1  1  1 
1  11 

222 

14 
15 
IG 

13S0 
1413 
1445 

1384  1387  1390 
1416  1419  1422 
1449  1452  1455 

1393  1396  1400 
1426  1429  1432 
1459  1462  1466 

1403 
1435 
1469 

1406  1409 
1439  1442 
1472  1476 

4 
3 
3 

223 
334 
344 

17 

18 
19 

1479 
1514 
1549 

1483  1486  1489 
1517  1521  1524 
1552  1556  1560 

1493  1496 
1528  1531 
1563  1567 

1500 
1535 
1570 

1503 
1538 
1574 

1507  1510 
1542  1545 
1578  1581 

4 

4 
4 

445 
456 
556 

20 

15S5 

1589  1592  1596 

1600  1603 

1607 

1611 

1614  1618 

4 

8  9 

21 
23 

1622 
1660 
1698 

1626  1629  1633 
1663  1667  1671 
1702  1706  1710 

1637  1641  1644 
1675  1679  1683 
1714  1718  1722 

1648  1652  1656 
1687  1690  1694 
1726  1730  1734 

4 
4 
4 

1  1 

2  2 
2  3 

24 
25 
26 

1738 
1778 
1820 

1742  1746  1750 
1782  1786  1791 
1824  1828  1832 

1754  1758 
1795  1799 
1837  1841 

1762 
1803 
1845 

1766  1770  1774 
1807  1811  1816 
1849  1854  1858 

4 
4 
4 

3  4 

4  5 

5  5 

27 

28 
29 

1862 
1905 
1950 

1866  1871  1875 
1910  1914  1919 
1954  1959  1963 

1879  1884 
1923  1928 
1968  1972 

1888 
1932 
1977 

1892 
1936 
1982 

1897  1901 
1941  1945 
1986  1991 

4 

5 
4 

6  6 

6  7 

7  8 

30 

1995 

2000  2004  2009 

2014  2018  2023 

2028  2032  2037 

5 

1011 

31 
32 
33 

2042 
2089 
2138 

2046  2051  2056 
2094  2099  2104 
2143  2148  2153 

2061  2065 
2109  2113 
2158  2163 

2070 
2118 
2168 

2075 
2123 
2173 

2080  2084 
2128  2133 
2178  2183 

5 
5 
5 

1  1 

2  2 

3  3 

34 
35 
30 

2188 
2239 
2291 

2193  2198  2203 
2244  2249  2254 
2296  2301  2307 

2208  2213 
2259  2265 
2312  2317 

2218 
2270 
2323 

2223 
2275 
2328 

2228  2234 
2280  2286 
2333  2339 

5 
5 
5 

4  4 

5  6 

6  7 

37 
38 
39 

2344 
2399 
2455 

2350  2355  2360 
2404  2410  2415 
2460  2466  2472 

2366  2371 
2421  2427 
2477  2-183 

2377 
2432 
2^89 

2382 
2438 
2495 

2388  2393 
2443  2449 
2500  2506 

6 
6 
6 

7  8 

8  9 

9  10 

40 

2512 

2518  2523  2529 

2535  2541 

2547 

2553 

2559  2564 

6 

1213 

41 
42 
43 

2570 
2630 
2692 

2576  2582  2588 
2636  2642  2649 
2698  2704  2710 

2594  2600  2606 
2655  2661  2667 
2716  2723  2729 

2612 
2673 
2735 

2618  2624 
2679  2685 
2742  2748 

6 

7 
6 

1  1 

2  3 
4  4 

44 
45 
40 

2754 
2818 
2884 

2761  2767  2773 
2825  2831  2838 
2891  2897  2904 

2780  2786  2793 
28 H  2851  2858 
2911  2917  2924 

2799  2S05  2812 
2864  2871  2877 
2931  2938  2944 

6 

5  5 

6  7 

7  8 

47  2951 

48  3020 

49  ,  3090 

2958  2965  2972 
3027  3034  3041 
3097  3105  3112 

2979  2985  2992 
3048  3055  3062 
3119  3126  3133 

2999  3006  3013 
3069  3076  3083 
3141  3148  3155 

8  9 

10  10 

11  12 

10 


11.   ANTILOGAEITHMS. 


n 

0 

12    3 

4   5   6 

7    8   9 

d 

pp. 

50 

3162 

3170  3177  3184 

3192  3199  3206 

3214  3221  3228 

8 

1415 

51 
52 
53 

3236 
3311 

3388 

3243  3251  3258 
3319  3327  3334 
3396  3404  3412 

3266  3273  3281 
3342  3350  3357 
3420  3428  3436 

3289  3296  3304 
3365  3373  3381 
3443  3451  3459 

7 
7 
8 

1  2 

3  3 

4  5 

54 
55 
56 

3467 
3548 
3631 

3475  3483  3491 
3556  3565  3573 
3639  3648  3656 

3499  3508  3516 
3581  3589  3597 
3664  3673  3681 

3524  3532  3540 
3606  3614  3622 
3690  3698  3707 

8 
9 
8 

6  6 

7  8 

8  9 

57 

58 
59 

60 

3715 
3802 
3S90 

3724  3733  3741 
3811  3819  3828 
3899  3908  3917 

3750  3758  3767 
3837  3846  3855 
3926  3936  3945 

3776  3784  3793 
3864  3873  3882 
3954  3963  3972 

9 
8 
9 

1011 
1112 
13  14 

3981 

3990  3999  4009 

4018  4027  4036 

4046  4055  4064 

lo!l617| 

61 
62 
63 

4074 
4169 
4266 

4083  4093  4102 
4178  41SS  4198 
4276  4285  4295 

4111  4121  4130 
4207  4217  4227 
4305  4315  4325 

4140  4150  4159 
4236  4246  4256 
4335  4345  4355 

10 
10 
10 

11 

5  5 

64 
65 
66 

4365 
4467 
4571 

4375  4385  4395 
4477  4487  4498 
4581  4592  4603 

4406  4416  4426 
4508  4519  4529 
4613  4624  4634 

4436  4446  4457 
4539  4550  4560 
4645  4656  4667 

10 

11 

10 

6  7 
8  9 
1010 

67 
68 
69 

4677 
4786 
4898 

4688  4699  4710 
4797  4808  4819 
4909  4920  4932 

4721  4732  4742 
4831  4842  4853 
4943  4955  4966 

4753  4764  4775 
4864  4875  4887 
4977  4989  5000 

11 
11 

12 

1112 
13  14 
1415 

70 

5012 

5023  5035  5047 

5058  5070  5082 

5093  5105  5117 

12|l819| 

71 
72 
73 

5129 

5248 
5370 

5140  5152  5164 
5260  5272  5284 
5383  5395  5408 

5176  5188  5200 
5297  5309  5321 
5420  5433  5445 

5212  5224  5236 
5333  5346  5358 
5458  5470  5483 

12 
12 
12 

2  2 

4  4 

5  6 

74 
75 
76 

5495 
S623 
5754 

5508  5521  5534 
5636  5649  5662 
5768  5781  5794 

5546  5559  5572 
5675  5689  5702 
5808  5821  5834 

5585  5598  5610 
5715  5728  5741 
5848  5861  5875 

13 
13 
13 

7  8 
9  10 

nil 

77 
78 
79 

5888 
6026 
6166 

5902  5916  5929 
6039  6053  6067 
6180  6194  6209 

5943  5957  5970 
6081  6095  6109 
6223  6237  6252 

5984  5998  6012 
6124  6138  6152 
6266  6281  6295 

14 
14 
15 

13  13 
1415 
1617 

80 

6310 

6324  6339  6353  !  6368  6383  6397 

6412  6427  6442 

15  202l| 

81 
82 
83 

6457 
6607 
6761 

6471  6486  6501 
6622  6637  6653 
6776  6792  6S0S 

6516  6531  6546 
6668  6683  6699 
6823  6839  6855 

6561  6577  6592 
6714  6730  6745 
6871  6887  6902 

15 
16 
16 

2  2 
4  4 
6  6 

84 
85 
86 

6918 
7079 
7244 

6934  6950  6966 
7096  7112  7129 
7261  7278  7295 

6982  6998  7015 
7145  7161  7178 
7311  7328  7345 

7031  7047  7063 
7194  7211  7228 
7362  7379  7396 

16 
16 
17 

8  8 
1011 
12  13 

87 
88 
89 

7413 
7586 
7762 

7430  7447  7464 
7603  7621  7638 
7780  7798  7816 

7482  7499  7516 
7656  7674  7691 
7834  7852  7870 

7534  7551  7568 
7709  7727  7745 
7889  7907  7925 

IS 
17 
IS 

14  15 
1617 
1819 

90 

7943 

7962  7980  7998  8017  8035  8054 

8072  8091  8110 

18 

2223 

91 
92 
93 

8128 
8318 
8511 

8147  8166  8185 
8337  8356  8375 
8531  8551  8570 

8204  8222  8241 
8395  8414  8433 
8590  8610  8630 

8260  8279  8299 
8453  8472  8492 
8650  8670  8690 

19 
19 
20 

2  2 
4  5 

7  7 

94 
95 
96 

8710 
8913 
9120 

S730  8750  8770 
8933  8954  8974 
9141  9162  9183 

8790  8810  8831 
8995  9016  9036 
9204  9226  9247 

8851  8872  SS92 
9057  9078  9099 
9268  9290  9311 

21 
21 
22 

9  9 
1112 
13  14 

97 

98 
99 

9333 
9550 
9772 

9354  9376  9397 
9572  9594  9616 
9795  9817  9840 

94-19  9441  9462 
9638  9661  9683 
9863  9886  9908 

9484  9506  9528 
9705  9727  9750 
9931  9954  9977 

22 
22 
23 

15  16 
1818 
20  21 

III.   ADDITION  LOGAKITHMS.       log(l  +  n) 


li^ 

0 

1   2   3  1  4   5   « 

7    8 

9 

d 

pp. 

«.  o.uuuo 

0001  0001  0001 

0001  0001  0002 

0002  0003  0003 

1 

7.0 

1 

2 

3 

U.0004  OOCH  OOOi  0005 
0005'  0006  0006  0006 
0007  0007  0007  0007 
0009  0009  0009  0009 

0005  0005  0005 

0006  0006  0006 

0008  0008  0008 

0009  0010  0010 

0005  0005  0005 

0006  0007  0007 
0008  0008  0008 
0010  0010  0011 

0 
0 

1 
0 

12 

00 
00 
01 

4 
5 

« 

0011  0011  0011  0012 
0014  0014  0014  0015 
0017|  0018  0018  0018 

0012  0012  0013 
0015  0015  0016 
0019  0019  0020 

0013  0013 
0016  0016 
0020  0021 

0013 
0017 
0021 

1 

0 

1 

01 

1 1 
1 1 

7 
8 
9 

0022 
0027 
0034 

0022  0023  0023 
0028  0029  0029 
0035  0036  0037 

0024  0024  0025 
0030  0031  0031 
0038  0039  0039 

0025  0026 
0032  0033 
0040  0041 

0027 
C034 

0042 

0 
0 

1 

1 1 

12 
1  2 
34 

8.0  0.0043 

0044  0045  0046 

0047  0048  0050 

0051  0052 

0053 

1 

1  0054]  0056  0057  0058 

2  '  006S,  0070  0071  0073 

3  0086  0088  0090  0092 

0060  0061  0062 
0075  0077  0078 
0094  0096  0098 

0064  0065 
0080  0082 
0101  0103 

0067 
0084 
0105 

1 

2 
3 

00 
1  1 
11 

4  i  0108 

5  0135 
(J   0170 

0110  0113  0115 
0138  0141  0145 
0173  0177  0181 

0118  0121  0123 
0148  0151  0155 
0186  0190  0194 

0126  0129 
0158  0162 
0199  0203 

0132 
0166 
0208 

3 
4 
4 

1  2 
22 
22 

7 

0212 
0266 
0332 

0217  0222  0227 
0272  0278  0284 
0339  0347  0355 

0232  0238  0243 
0291  0297  0304 
0363  0371  0379 

0248  0254  0260 
0311  0318  0325 
0387  0396  0405 

6 

7 
9 

23 
23 
34 

9.0 

1 

0414 
0515 

0423  0432  0442 
0526  0538  0550 

0452  0462  0472 
0562  0574  0586 

0482  0493 
0599  0612 

0504 
0625 

11 

14 

9.20  0.0639 

0640  0642  0643 

0644  0646  0647 

0649  0650  0651 

2 

5(J 

21  06531  0654  0656  0657 

22  0667;  0668  0670  0671 

23  0681  0683  0684  0686 

0658  0660  0661 
0673  0674  0675 
06S7  0689  0690 

0663  0664 
0677  0678 
0691  0693 

0665 
0680 
0694 

2 
1 
2 

1  1 
1  1 

22 

24  0696  0697  0699  0700 

25  0711  0712  0714  0715 
20]  0726i  0728  0729  0731 

0702  0703  0705 
0717  07-18  0720 
0732  0^34  0735 

0706  0708  0709 
0721  0723  0725 
0737  0738  0740 

2 
1 
2 

22 
33 
34 

27 

28 
29 

0742 
0757 
0774 

0743  0745  0746 
0759  0761  0762 
0775  0777  0779 

0748  0749  0751 
0764  0766  0767 
0780  0782  0783 

0753  0754  0756 
0769  0770  0772 
0785  0787  0788 

1 
2 
2 

44 
45 
5  5 

9.30  0.0790  0792  0793  0795 

0797  0798  0800 

0802  0804  0805 

2 

78 

31 '  0807  0809  0810  0812 
32:  08241  0826  0827  0829 
33  0841  0843  OS45  0847 

0814  0815  0817 
0831  0833  0834 
0849  0850  0852 

0819  0821  0822 
0836  0838  0840 
0854  0856  0857 

2 

1 
2 

1  1 
12 

22 

34'  0859  0861  0863  0865 
35 1  0877:  0879  0881  0883 
3G'  0S96  0898  0900  0901 

0866  0868  0870 
0885  0887  0888 
0903  0905  0907 

0872  0874 
0890  0892 
0909  0911 

0876 
0894 
0913 

1 

2 
2 

33 

44 
45 

37 1  0915 
38  0934 
39:  0953 

0917  0918  0920 
0936  0938  0940 
0955  0957  0959 

0922  0924  0926 
0942  0944  0945 
0961  0963  0965 

0928  0930  0932 
0947  0949  0951 
0967  0969  0971 

2 
2 
2 

56 
66 
67 

9.40  0.U973 

0975  0977  0979 

0981  0983  0985 

0987  0989  0991 

2 

9 

41 
42 
43 

0993  0996  0998  1000 
1014  1016  leiS  1020 
1035 1  1037  1039  1042 

1002  1004  1006 
1022  1025  1027 
1044  1046  1048 

lOOS  1010 
1029  1031 
1050  1052 

1012 
1033 
1054 

2 
2 
3 

1 
2 
3 

44 
45 
4G 

10571  1059  1061  1063 
1078  1081  1083  1085 
1101  1103  1105  1107 

1065  1067  1070 
1087  1089  1092 
1110  1112  1114 

1072  1074 
1094  1096 
1116  1119 

1076 
1098 
1121 

2 
3 
2 

4 

5 
5 

47|  1123]  1125  1128  1130 

48  11461  1148  1151  1153 

49  1169!  1172  1174  1177 

1132  1135  1137 
1155  1158  1160 
1179  1181  1184 

1139  1141 
1162  1165 
1186  1189 

1144 
1167 
1191 

2 
2 
2 

6 

7 
8 

log(l  +  n)      III.   ADDITION  LOGAEITHMS. 


log/? 

0 

1 

2   3 

4   5   6 

7 

8   9 

d 

3 

pp. 

9.50 

0.1193 

1196  1198  1201 

1203  1205  1208 

1210 

1213  1215 

10 

51 
52 
53 

1218 
1242 
1267 

1220  1222  1225 
1245  1247  1250 
1270  1272  1275 

1227  1230  1232 
1252  1255  1257 
1277  1280  1283 

1235  1237  1240 
1260  1262  1265 
1285  1288  1290 

2 
2 
3 

1 
2 
3 

54 
55 
56 

1293 
1319 
1345 

1295 
1321 
1348 

1298  1301 
1324  1327 
1351  1353 

1303  1306  1308 
1329  1332  1335 
1356  1359  1361 

1311 
1337 
1364 

1314  1316 
1340  1343 
1367  1369 

3 

2 
3 

4 
5 
6 

57 

58 
59 

1372 
1399 
1427 

1375  1377  1380 
1402  1405  1408 
1430  1433  1436 

1383  1386  1388 
1410  1413  1416 
1438  1441  1444 

1391 
1419 
1447 

1394  1397 
1422  1424 
1450  1453 

2 
3 

2 

3 

7 
8 
9 

9.60 

0.1455 

1458 

1461  1464 

1467  1470  1473 

1475 

1478  1481 

11 

61 
62 
63 

14S4 
1513 
1543 

1487 
1516 
1546 

1490  1493 
1519  1522 
1549  1552 

1496  1499  1502 
1525  1528  1531 
1555  1558  1561 

1504 
1534 
1564 

1507  1510 
1537  1540 
1567  1570 

3 
3 
3 

1 
2 
3 

M 
65 
66 

1573 
1604 
1635 

1576 
1607 
1638 

1579  1582 
1610  1613 
1641  1644 

1585  1588  1591 
1616  1619  1622 
1647  1651  1654 

1594 
1625 
1657 

1598  1601 
1629  1632 
1660  1663 

3 
3 
3 

4 
6 

7 

67 
68 
69 

9.70 

1666 
1699 
1731 

0.1764 

1670 
1702 
1735 

1673  1676 
1705  1708 
1738  1741 

1679  1682  1686 
1712  1715  1718 
1744  1748  1751 

1689 
1721 
1754 

1692  1695 
1725  1728 
1758  1761 

4 
3 
3 

7 

8 
9 
10 

1768 

1771  1774 

1778  1781  178f 

1788 

1791  1795 

12 

71 
72 
73 

1798 
1832 
1867 

1801 
1836 
1870 

1805  1808 
1839  1842 
1874  1877 

1812  1815  1818 
1846  1849  1853 
1881  1884  1888 

1822 
1856 
1891 

1825  1829 
1860  1863 
1895  1898 

3 

4 
4 

1 
2 
4 

74 
75 

76 

1902 
1938 
1974 

1906 
1941 
1978 

1909  1913 
1945  1949 
1981  1985 

1916  1920  1923 
1952  1956  1959 
1989  1992  1996 

1927  1931  1934 
1963  1967  1970 
2000  2003  2007 

4 
4 
4 

5 
6 

7 

77 
78 
79 

2011 
2048 
2086 

2015 
2052 
2090 

2018  2022 
2056  2059 
2094  2097 

2026  2029  2033 
2063  2067  2071 
2101  2105  2109 

2037 
2075 
2113 

2041  2044 
2078  2082 
2117  2121 

4 
4 
3 

4 

8 
10 
11 

9.80 

0.2124 

2128  2132  2136 

2140  2144  2148 

2152 

2156  2159 

13 

81 

82 
83 

2163 
2203 
2243 

2167 

2207 
2247 

2171  2175 
2211  2215 
2251  2255 

2179  2183  2187 
2219  2223  2227 
2259  2263  2267 

2191 
2231 

2271 

2195  2199 
2235  2239 
2275  2279 

4 
4 

5 

1 
3 
4 

84 
85 
86 

2284 
2325 
2366 

2288 
2329 
2371 

2292  2296 
2333  2337 
2375  2379 

2300  2304  2308 
2341  2346  2350 
2383  2388  2392 

2312  2316  2321 
2354  2358  2362 
2396  2400  2405 

4 
4 
4 

5 

7 
8 

87 
88 
89 

9.90 

2409 
2452 
2495 

2413  2417  2422 
2456  2460  2465 
2499  2504  2508 

2426  2430  2434 
2469  2473  2478 
2513  2517  2521 

2439  2443  2447 
2482  2486  2491 
2526  2530  2535 

5 
4 
4 

5 

9 
10 
12 

0.2539 

2543 

2548  2552 

2557  2561  2566 

2570  2575  2579 

14 

91 
92 
93 

2584 
2629 
2674 

2588  2593  2597 
2633  2638  2642 
2679  2684  2688 

2602  2606  2611 
2647  2651  2656 
2693  2697  2702 

2615 
2661 
2707 

2620  2624 
2665  2670 
2711  2716 

5 
4 

5 

1 
3 
4 

94 
95 
96 

2721 
2767 
2815 

2725  2730  2735 
2772  2777  2782 
2820  2824  2829 

2739  2744  2749 
2786  2791  2796 
2834  2839  2844 

2753  2758  2763 
2801  2805  2810 
2848  2853  2S5S 

4 

5 
5 

6 

7 
8 

97 
98 
99 

2863 
2911 
2961 

2868  2873  2877 
2916  2921  2926 
2966  2970  2975 

2882  2887  2892 
2931  2936  2941 
2980  2985  2990 

2897  2902  2907 
2946  2951  2956 
2995  3000  3005 

4 
5 

5 

10 
11 
13 

10 


IV.     SUBTEAOTION  LOGAKITHMS.       log 


logr/i 

0 

1   2   3 

0001  0001  0001 

4   5   6 

0001  0001  0002 

7    8   9 

d 

1 

pp. 

6.   0.0000 

0002  0003  0003 

7.0  0.0004  OOCH  OOOi  OOOi 

0005  0005  0005 

0005  0005  0005 

0 

1  2 

1 
2 
3 

0005 
0007 

ooo<; 

0006  0006  0006 

0007  0007  0007 
0009  0009  0009 

0006  0006  0006 
0008  0008  0008 
0010  0010  0010 

0006  0007  0007 
0008  0008  0008 
0010  0010  0011 

0 

1 

0 

0  0 
0  0 
0  1 

4 
5 
6 

0011 
0014 
0017 

0011  0011  0012 
0014  0014  0015 
0018  OOIS  0019 

0012  0012  0013 
0015  0015  0016 
0019  0019  0020 

0013  0013  0013 
0016  0017  0017 
0020  0021  0021 

1 

0 

1 

0  1 

1  1 
1  1 

7 
8 
9 

0022 
0027 
0035 

0022  0023  0023 
0028  0029  0029 
0035  0036  0037 

0024  0024  0025 
0030  0031  0032 
0038  0039  0040 

0026  0026  0027 
0032  0033  0034 
0041  0042  0043 

0 

1 
1 

1  1 

1  2 
1  2 

8.0  0.0044 

0045  0046  0047 

0048  0049  0050 

0051  0053  0054 

1 

3  4 

1 
2 
3 

0055 
0069 
0088 

0056  0058  0059 
0071  0073  0074 
0090  0092  0094 

0060  0062  0063 
0076  0078  0080 
0096  0098  0101 

0065  0066  0068 
0082  0084  0086 
0103  0105  0108 

1 

2 

2 

0  0 

1  1 
1  1 

4 
5 
6 

0110 
0140 
0176 

0113  0116  0118 
0143  0146  0150 
0181  0185  0189 

0121  0124  0127 
0153  0157  0161 
0194  0198  0203 

0130  0133  0136 
0164  0168  0172 
0208  0213  0218 

4 
4 

5 

1  2 

2  2 
2  2 

7 
8 
9 

0223 
0283 
0359 

0229  0234  0240 
0290  0297  0304 
0368  0377  0386 

0245  0251  0257 
0311  0319  0327 
0396  0405  0415 

0264  027Q  0276 
0335  0343  0351 
0425  0436  0447 

7 
8 
11 
14 

18 

2  3 

2  3 

3  4 

9.0 

1 

0458 
0584 

0469  0480  0492 
0599  0614  0629 

0504  0517  0530 
0645  0661  0678 

0543  0556  0570 
0695  0713  0731 

9.20  0.0749 

0751  0753  0755 

0757  0759  0761 

0763  0765  0767 

1 

56 

21  0768 
22;  078S 

23  0808 

0770  0772  0774 
0790  0792  0794 
0810  0812  0814 

0776  0778  0780 
0796  0798  0800 
0817  0819  0821 

0782  0784  0786 
0802  0804  0806 
0823  0825  0827 

2 
2 
2 

1  1 

1  1 

2  2 

24  0829 
25 1  OS.SO 
26  0872 

0831  0833  0835 
0853  0855  0857 
0875  0877  0879 

0838  0840  0842 
0859  0861  0863 
0881  0884  0886 

0844  0846  0848 
0866  0868  0870 
0888  0890  0893 

2 

2 

2 

2  2 

3  3 
3  4 

27  0895 

28  0918 
|29  0942 

0897  0899  0902 
0920  0923  0925 
0944  0947  0949 

0904  0906  0909 
0928  0930  0932 
0952  0954  0957 

0911  0913  0916 
0935  0937  0940 
0959  0962  0964 

2 

2 

3 
3 

4  4 

4  5 

5  5 

9.30  0.0967 

0969  0972  0974 

0977  0979  0982 

0984  0987  0989 

7  8 

31  0992 

32  1018 
33j  1045 

0994  0997  1000 
1020  1023  1026 
1047  1050  1053 

1002  1005  1007 
1028  1031  1034 
1056  1058  1061 

1010  1013  1015 
1037  1039  1042 
1064  1067  1069 

3 
3 
3 

1  1 

1  2 

2  2 

34  1072 

35  1101 

36  1130 

1075  1078  1081 
1104  1106  1109 
1133  1136  1139 

1084  1086  1089 
1112  1115  1118 
1142  1145  1148 

1092  1095  1098 
1121  1124  1127 
1151  1154  1157 

3 
3 
3 

3  3 

4  4 
4  5 

37 
38 

39 

1160 
1191 
1223 

1163  1166  1169 
1194  1198  1201 
1226  1230  1233 

1172  1176  1179 
1204  1207  1210 
1236  1240  1243 

1182  1185  1188 
1214  1217  1220 
1246  1250  1253 

3 
3 
3 
3 

5  6 

6  6 

6  7 

9.40  0.1256 

1260  1263  1266 

1270  1273  1277 

1280  1283  1287 

9  11 

41  1290 

42  1325 

43  1362 

1294  1297  1301 
1329  1333  1336 
1365  1369  1373 

1304  1308  1311 
1340  1343  1347 
1377  1380  1384 

1315  1318  1322 
1351  1354  1358 
1388  1392  1395 

3 

4 
4 

1  1 

2  2 

3  3 

44  1399 

45  1438 

46  1478 

1403  1407  1411 
1442  1446  1450 
1482  1486  1490 

1414  1418  1422 
1454  1458  1462 
1494  1498  1502 

1426  1430  1434 
1466  1470  1474 
1506  1511  1515 

4 
4 
4 

4  4 

5  6 

5  7 

47  1519 
48|  1561 
49  1605 

1523  1527  1531 
1566  1570  1574 
1610  1614  1619 

1536  1540  1544 
1579  15S3  1588 
1623  1628  1633 

1549  15.53  1.557 
1592  1596  1601 
1637  1642  1646 

1 

5 

6  8 

7  9 

8  10 

log  I 


IV.   SUBTKAOTION  LOGAEITHMS. 


11 


log /I 

0 

12    3 

4   5   6 

7    8   9 

a  pp. 

9.50 

0.1651 

1656  1660  1665 

1670  1674  1679 

1684  1688  1693 

5  1213 

51 
52 
53 

1698 
1747 
1797 

1703  1708  1712 
1752  1757  1762 
1802  1807  1812 

1717  1722  1727 
1767  1772  1777 
1818  1823  1828 

1732  1737  1742 
1782  1787  1792 
1833  1839  1844 

5  1  1 
5  2  3 
5  4  4 

54 
55 

1849 
1903 
1959 

1854  1860  1865 
1909  1914  1920 
1965  1971  1976 

1871  1876  1881 
1925  1931  1936 
1982  1988  1994 

1887  1892  1898 
1942  1948  1953  ( 
2000  2005  2011  ( 

5  5  5 
3  6  7 
S  7  8 

57 

58 
59 

2017 
2077 
2140 

2023  2029  2035 
2084  2090  2096 
2146  2153  2159 

2041  2047  2053 
2102  2108  2115 
2166  2172  2179 

2059  2065  2071  ( 
2121  2127  2134  ( 
2185  2192  2198  ' 

■->    8  9 
S  1010 
7    1112 

9.60 

0.2205 

2211  2218  2225 

2231  2238  2245 

2252  2259  2265  ' 

J   1415 

61 
62 
63 

2272 
2342 
2415 

2279  2286  2293 
2350  2357  2364 
2423  2430  2438 

2300  2307  2314 
2371  2378  2386 
2445  2453  2460 

2321  2328  2335  ' 
2393  2400  2408  ' 
2468  2476  2483  i 

J    1  2 
J    3  3 
5  4  5 

64 
65 
66 

2491 

2570 
2653 

2499  2507  2515 
2578  2586  2595 
2661  2670  2678 

2522  2530  2538 
2603  2611  2619 
2687  2695  2704 

2546  2554  2562  { 
2628  2636  2644  ' 
2713  2721  2730  « 

I    6  6 

}    7  8 
}    8  9 

67 

68 
69 

2739 
2829 
2922 

2748  2756  2765 
2838  2847  2856 
2932  2942  2951 

2774  2783  2792 
2866  2S75  2884 
2961  297i  2981 

2801  2810  2819  1 
2894  2903  2913  < 
2991  3001  3011  1 

0  1011 
}    1112 
0  1314 

9.70 

0.3021 

3031  3041  3051 

3061  3071  3082 

3092  3103  3113  ] 

0  1617 

71 
72 
73 

3123 
3231 
3344 

3134  3145  3155 
3242  3253  3265 
3356  3368  3380 

3166  3177  3188 
3276  3287  3299 
3391  3403  3415 

3 198  3209  3220  1 
3310  3321  3333  1 
3427  3439  3451  1 

1  2  2 

1  3  3 

2  5  5 

74 
75 
76 

3463 
3589 
3721 

3476  3488  3500 
3602  3614  3627 
3734  3748  3762 

3513  3525  3538 
3641  3654  3667 
3775  3789  3803 

3550  3563  3576  1 
3680  3694  3707  1 
3817  3831  3846  1 

3  6  7 

4  8  9 
4  1010 

77 
78 
79 

9.80 

3860 
4007 
4163 

3874  3889  3903 
4022  4038  4053 
4180  4196  4212 

3918  3933  3947 
4069  4084  4100 
4229  4245  4262 

3962  3977  3992  1 
4116  4131  4147  1 
4278  4295  4312  1 

5  1112 

6  1314 

7  1415 

0.4329 

4346  4364  4381 

4398  4416  4434 

4452  4470  4488  1 

8  181t 

81 
82 
83 

4506 
4694 
4896 

4524  4542  4561 
4714  4733  4753 
4917  4938  4959 

45S0  4598  4617 
4773  4793  4813 
4980  5002  5024 

4636  4655  4675  1 
4834  4854  4875  2 
5046  5068  5090  2 

9  2  2. 

1  4  4 

2  5  6 

84 
85 
86 

5112 
5345 
5598 

5135  5157  5180 
5370  5394  5419 
5624  5651  5678 

5203  5227  5250 
5444  5469  5494 
5705  5732  5760 

5273  5297  5321  2 
5520  5546  5572  2 
5787  5815  5844  2 

4  7  8 
6  9  10 
8  1111 

87 
88 
89 

5872 
6172 
6502 

5901  5930  5959 
6204  6236  6268 
6537  6572  6608 

5989  6019  6049 
6300  6333  6366 
6644  6680  6717 

6079  6110  6141  3 
6400  6434  6468  3 
6754  6792  6830  3 

1  1313 

4  1415 

5  1617 

pp. 

1 
2 
3 

20  21  22  23 

24  25  26  1  27  28  29  ■  30 

3132  33  34  35  36,  37  38 

2 
4 
6 

2  2  2 
4  4  5 
6  7  7 

2  3  3 

5  5  5 
7  8  8 

3  3  3 
5  6  6 
8  8  9 

3 
6 
9 

3  3  3 
6  6  7 
9  10  10 

3  4  4 

7  7  7 

10  11  11 

4  4 

7  8 

11  11 

4 
5 
6 

8 
10 
12 

8  9  9 
11  11  12 
13  13  14 

10  10  10 
12  13  13 
14  15  16 

11  11  12 
14  14  15 
16  17  17 

12 
15 
18 

12  13  13 
16  16  17 
19  19  20 

14  14  14 
17  18  18 
20  21  22 

15  15 
19  19 
22  23 

7 
8 
9 

14 
16 
18 

15  15  16 
17  18  18 
19  20  21 

17  18  18 
19  20  21 
22  23  23 

19  20  20 
22  22  23 
24  25  26 

21 
24 

27 

22  22  23 

25  26  26 
28  29  30 

24  25  25 
27  28  29 
31  32  32 

26  27 
30  30 
ii   34 

12 


IV.   SUBTEAOTION  LOGARITHMS 


'«^(r^) 


logn 

0 

12   3   4   5   6 

7    8    9 

d 

pp. 

9.9(K)  0.6S6S 

6872  6876  6880  6884  6888  6892 

6895  6899  6903 

4  5 

901  6907 
9(»2  6946 
903 1  6986 

6911  6915  6919 
6950  6954  6958 
6990  6994  6998 

6923  6927  6931 
6962  6966  6970 
7002  7006  7010 

6935  6938  6942 
6974  6978  6982 
7014  7018  7022 

0  1 

1  1 
1  2 

904 
905 
906 

7026  7030  7034  7038 
7067  7071  7075  7079 
7108  7112  7116  7121 

7042  7047  7051 
7083  7087  7092 
7125  7129  7133 

7055  7059  7063 
7096  7100  7104 
7137  7141  7146 

2  2 
2  3 
2  3 

907 
908 
909 

7150  7154  7158  7162 
7192  7196  7200  7205 
7234  7239  7243  7247 

7167  7171  7175 
7209  7213  7217 
7252  7256  7260 

7179  7183  7188 
7222  7226  7230 
7265  7269  7273 

— 

3  4 

3  4 

4  5 

9.910  0.7278  17282  7286  7291 

7295  7299  7304 

7308  7313  7317 

6  7 

911 
912 
913 

7321 
7366 
7410 

7326  7330  7335 
7370  7375  7379 
7415  7419  7424 

7339  7343  7348 
7383  7388  7392 
7428  7433  7438 

7352  7357  7361 
7397  7401  7406 
7442  7447  7451 

1  1 

1  1 

2  2 

914 
915 
916 

7456 
7502 
7548 

7460  7465  7469 
7506  7511  7516 
7553  7558  7562 

7474  7479  7483 
7520  7525  7530 
7567  7572  7577 

7488  7492  7497 
7534  7539  7544 
7581  7586  7591 

2  3 

3  4 

4  4 

917 
918 
919 

7595 
7643 
7692 

7600  7605  7610 
7648  7653  7658 
7697  7701  7706 

7614  7619  7624 
7663  7667  7672 
7711  7716  7721 

7629  7634  7638 
7677  7682  7687 
7726  7731  7736 

— 

4  5 

5  6 
5  6 

9.920  0.7741 

7746  7751  7756 

7761  7766  7771 

7776  7781  7786 

8  9 

921  7791 

922  7841 

923  7892 

7796  7801  7806 
7846  7851  7856 
7897  7903  7908 

7811  7816  7821 
7861  7867  7872 
7913  7918  7923 

7826  7831  7836 
7877  7882  7887 
7929  7934  7939 

5 

1  1 

2  2 
2  3 

924 
925 
926 

7944 
7997 
8050 

7949  7955  7960 
8002  8007  8013 
8056  8061  8066 

7965  7970  7976 
8018  8023  8029 
8072  8077  8083 

7981  7986  7992 
8034  8040  8045 
8088  8094  8099 

6 

3  4 

4  5 

5  5 

927 
928 
929 

8105 
8160 
8215 

8110  8115  8121 
8165  8171  8176 
8221  8227  8232 

8126  8132  8137  I  8143  8148  8154 
8182  8187  8193  8199  8204  8210 
8238  8244  8249  ,  8255  8261  8266 

6 

5 
6 

6 

6  6 

6  7 

7  8 

9.93010.8272 

8278  8284  8289 

8295  8301  8307 

8312  8318  8324 

1011 

931  8330 

932  83S8 

933  8448 

8336  8341  8347 
8394  8400  8406 
8154  8460  8466 

8353  8359  8365 
8412  8418  8424 
8472  8478  8484 

8371  8377  8382 
8430  8436  8442 
8490  8496  8502 

6 
6 
6 

1  1 

2  2 

3  3 

934  8508 
935'  8570 
936;  8632 

8514  8520  8527 
8576  8582  8588 
8638  8645  8651 

8533  8539  8545 
8595  8601  8607 
8657  8664  8670 

8551  8557  8563 
8613  8620  8626 
8676  8683  8689 

7 
6 
7 

4  4 

5  6 

6  7 

937 1  8696  18702  8708  8715 
938 1  8760  !  8767  8773  8780 
939  8826  8833  8839  SS46 

8721  8728  8734 
8786  8793  8800 
8853  8859  8866 

8741  8747  8754 
8806  8813  8819 
8873  8879  8886 

6 

7 
7 

7 

7  8 

8  9 

9  10 

9.9400.8893  8900  8906  8913 

8920  8927  8934 

8940  8947  8954 

1213 

941  8961 

942  9030 
943j  9101 

8968  8975  8982  !  8989  8996  9002 
9037  9044  9051  9058  9065  9073 
9108  9115  9122  ;  9130  9137  9144 

9009  9016  9023 
9080  9087  9094 
9151  9158  9166 

7 
7 
7 

1  1 

2  3 
4  4 

944|  9173 
945  9246 
946,  9321 

9180  9188  9195 
9254  9261  9269 
9329  9336  9344 

9202  9209  9217 
9276  9284  9291 
9351  9359  9367 

9224  9232  9239 
9299  9306  9314 
9374  9382  9390 

7 
7 
7 

5  5 

6  7 

7  8 

947  9397 
948 1  9475 
949  9555 

9405  9413  9421 
9483  9491  9499 
9563  9571  9579 

9428  9436  9444 
9507  9515  9523 
9587  9595  9603 

9452  9460  9467 
9531  9539  9547 
96119619  9628 

8 
8 
8 

8  9 

10  10 

11  12 

log( 

1 
1-n 

) 

CONTINUED. 

13 

logn       0 

1       2       3 

1    4       5       6 

7        8       9      (1 

pp. 
1415 

9.9o00.963f 

)   9644  9652  9660   9669  9677  9685 

9694  9702  9710   9 

951 
952 
953 

97]9:  9727  9735  9744    9752  9761  9769!  9778  9786  9795'  8 
9803.  9812  9820  9S29J  9838  9846  9S55{  9864  9872  9SS!!  9 
9890    9899  9907  9916'  9925  9934  9943    9951  9960  9969   9 

1    2 

3  3 

4  5 

954 
955 
956 

9978'  9987  9996*0005*0014*0023*0032*0041*0050*00601  9 

1.0069'  0078  0087  0096i  0106  0115  01241  0133  0143  01521  9 

0161    01710180  0190:  0199  0209  0218    0228  0237  0247i  9 

6  6 

7  8 

8  9 

957 
958 
959 

0256    0266  0276  0285    0295  0305  0314 
0354    0364  0373  0383    0393  0403  0413 
0453|  0464  0474  0484    0494  0504  0514 

0324  0334  0344  10 
0423  0433  0443  10 
0525  0535  0545  11 

10  11 
1112 
13  14 

1617 

9.9601.05561  0566  0576  05S7I  0597  060S  0618 

0629  0640  0650  11 

961 
962 
963 

0661:  06710682  0693!  0704  0714  0725 
0769'  0780  0791  0S02|  0813  0824  0835 
0S80|  08910902  0913 1  0925  0936  0948 

0736  0747  075s'll 
0846  0857  0868  12 
0959  0971  0982  12 

11 

5    5 

964 
965 
966 

0994 
1111 

1232 

1005  1017  1028    1040  1052  1064    1075  1087  109912 
1123  1135  1147,   1159  1171  1183!  1195  1207  1220  12 
1244  1257  1269    1281  1294  1306!  1319  1331  1344  13 

6   7 
8   9 
1010 

967 
968 
969 

1357 
1485 
1618 

13691382  1395    1408  1420  1433 
14981512  1525!  1538  1551  1565 
1632  1645  1659    1673  1686  1700 

1446  1459  1472  13 
1578  1591  1605  13 
1714  1728  1742  14 

1112 

13  14 

14  15 

1819 

9.970 

1.1756 

17701784  1798    1S12  1826  1841 

1855  1869  1884  14 

971 
972 
973 

1898;  1913  1927  1942:   1956  1971  1986    2001  2016  203115 
2046:  20612076  2091,  2106  2121  2137'  2152  2167  2183  16 
2199    2214  2230  22461  2261  2277  2293|  2309  2325  2341  16 

u 

974 
975 
976 

2357J  2374  2390  2406    2423  2439  24561  2473  2489  250617 
2523    2540  2557  2574    2591  2608  2625!  2643  2660  2678  17 
2695|  2713  2731  2748    2766  2784  28021  2820  2838  2857  18 

7    8 
910 
1111 

977 

978 
979 

2875    2893  2912  2931    2949  2968  2987 
3063    3082  3102  3121    3141  3161  3180 
3260^  3280  3301  3321    3342  3362  3383 

3006  3025  3044  19 
3200  3220  3240  20 
3404  3425  3446  21 

13  13 

14  15 
1617 

2021 

9.980 

1.3467    3488  3510  3531 

3553  3575  3596 

3618  3640  3663  22 

981 
982 
983 

3685    3707  3730  3753 
3915    3939  3962  3986 
4158:  4183  4208  4234 

3775  3798  3821 
4010  4035  4059 
4259  4285  4311 

3845  3868  3891 '24 
4084  4108  4133  25 
4337  4363  4390  26 

2    2 
4   4 
6    6 

984 
985 
986 

4416'  4443  4470  4497 
4692    4720  4749  4778 
49S6|  5017  5048  5079 

4524  4552  4579 
4807  4836  4866 
5110  5142  5174 

4607  4635  4663  29 
4896  4926  4956  30 
5206  5238  5270  33 

8   8 
1011 
12  13 

987 
988 
989 

5303 
5646 
6019 

5336  5370  5403 
5682  5718  5754 
6058  6097  6137 

5437  5471  5505 
5791  5828  5866 
6178  6218  6259 

5540  5575  5610  36 
5903  5942  5980  39 
6301  6343  6385,43 

1415 
1617 
1819 

2223 

9.990 

1.6428 

6471  6514  6559 

6603  6648  6693 

6739  6786  6833^47 

991 
992 
993 

6880'  6928  6977  7026 
7387 1  7441  7496  7551 
7962    8024  8087  8151 

7076  7126  7177 
7608  7665  7722 
8215  8281  8348 

7228  7281  7333  54 
7781  7840  7901  !61 
8416  8485  8555  71 

2    2 
4    5 
7    7 

994 
995 
996 

8626:' 

9413 

2.0377 

8699  8773  8848 
9500  9589  9680 
0487  0599  0714 

8924  9002  9081 
9773  9868  99651 
0833  0955  1080 

9162  9244  932S!85 
*0065*0166*0270i 
1209  1342  1480 

9   9 
1112 
13  14 

997 
998 
999 

1622 
3378 
6383 

1768  1920  2078 
3600  3834  4082 
6840  7351  7930 

2241  2411  2588 
4345  4624  4924 
8599  9391*0359| 

2772  2965  3166 
5245  5592  5969 
*1608*3369*6378 

15  16 
18  18 
20  21 

u 

V.  LOGARITHMIC  SINES 

AND  COSINES 

loir  sin  H                                 logrcosn^ 

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5640  5842  6035 
7330  7468  7602 

3880  4179  4459 
6220  6397  6567 
■7731  7857  7979 

4723  4971  5206 
6731  6889  7041 
8098  8213  8326 

5428 
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8436 

88 
87 
86 

4 
6 
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8436 

9403 

9.0192 

8543  8647  8749 
9489  9573  9655 
0264  0334  0403 

8849  89-16  9042 
9736  9816  9894 
0472  0539  060i 

9135  9226  9315 
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0670  0734  0797 

9403 

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84 
83 

88 
72 
62 

7 
8 
9 

0S59 
1436 
1943 

0920  0981  1040 
1489  1542  1594 
1991  2038  2085 

1099  1157  1214 
1646  1697  1747 
2131  2176  2221 

1271  1326-1381 
1797  1847  1895 
2266  2310  2353 

1436 
1943 
2397 

82 
81 
80 

^9 

55 
48 
44 

39 

To 

9.2397 

2439  2482  2524 

2565  2606  2647 

2687  2727  2767 

2806 

11 

12 

la 

2806 
3179 
3521 

2845  2883  2921 
3214  3250  3284 
3554  3586  3618 

2959  2997  3034 
3319  3353  3387 
3650  3682  3713 

3070  3107  3143 
3421  3455  3488 
374i  3775  3806 

3179 
3521 
3837 

78 
77 
76 

36 
33 
31 

14 
15 

1<> 

3837 
4130 
4403 

3867  3897  3927 
4158  4186  4214 
4430  4456  4482 

3957  3986  4015 
4242  4269  4296 
4508  4533  4559 

4044  4073  4102 
4323  4350  4377 
4584  4609  4634 

4130 
4403 
4659 

75 
74 
73 

28 
26 

25 

17 

18 
19 

4659 
4900 
5126 

4684  4709  4733 
4923  4946  4969 
5148  5170  5192 

4757  4781  4805 
4992  5015  5037 
5213  5235  5256 

4829  4853  4876 
5060  5082  5104 
5278  5299  5320 

4900 
5126 
5341 

72 
71 
70 

24 
22 
21 

20 

20 

9.5341 

5361  5382  5402 

5423  5443  5463 

5484  5504  5523 

5543 

69 

21 
22 
23 

5543 
5736 
5919 

5563  5583  5602 
57.54  5773  5792 
5937  5954  5972 

5621  5641  5660 
5810  5828  5847 
5990  6007  6024 

5679  5698  5717 
5865  5883  5901 
6042  6059  6076 

5736 
5919 
6093 

68 
67 
66 

19 
18 
17 

24 
25 
26 

6093 
6259 
6418 

6110  6127  6144 
6276  6292  6308 
6434  6449  6465 

6161  6177  6194 
6324  6340  6356 
6480  6495  6510 

6210  6227  6243 
6371  6387  6403 
6526  6541  6556 

6259 
6418 
6570 

65 
64 
63 

16 
15 
14 

27 

28 
29 

30 

6570 
6716 
6856 

6585  6600  6615 
6730  6744  6759 
6869  6883  6896 

6629  6644  6659 
6773  6787  6801 
6910  6923  6937 

6673  6687  6702 
6814  6828  6842 
6950  6963  6977 

6716 
6856 
6990 

62 
61 
60 

14 
14 
13 

12 

9.6990 

7003  7016  7029 

7042  7055  7068 

7080  7093  7106 

7118 

59 

31 
32 
33 

7118 
7242 
7361 

7131  7144  7156 
7254  7266  7278 
7373  7384  7396 

7168  7181  7193 
7290  7302  7314 
7407  7419  7430 

7205  7218  7230 
7326  7338  7349 
7442  7453  7464 

7242 
7361 
7476 

58 
57 
56 

12 
12 
12 

34 
35 
36 

7476 
7586 
7692 

7487  7-198  7509 
7597  7607  7618 
7703  7713  7723 

7520  7531  7542 
7629  7640  7650 
7734  7744  7754 

7553  7564  7575 
7661  7671  76S2 
7764  7774  7785 

7586 
7692 
7795 

55 
54 
53 

11 
10 
10 

37 
38 
39 

40 

7795 
7893 
7989 

7805  7815  7825 
7903  7913  7922 
7998  8007  8017 

7835  7844  7854 
7932  7941  7951 
8026  8035  8044 

7864  7874  7884 
7960  7970  7979 
8053  8063  8072 

7893 
7989 
8081 

52 
51 
50 

9 
10 
9 

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9.8081 

8090  8099  8108 

8117  8125  8134 

8143  8152  8161 

8169 

49 

41 
42 
43 

8169 
8255 
8338 

8178  8187  8195 
8264  8272  8280 
8346  8354  8362 

8204  8213  8221 
8289  8297  8305 
8370  8378  8386 

8230  8238  8247 
8313  8322  8330 
8394  8402  8410 

8255 
8338 
8418 

48 
47 
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8 
8 
8 

44 
45 
46 

8418 
8495 
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8426  8433  8441 
8502  8510  8517 
8577  8584  8591 

8449  8457  8464 
8525  8532  8540 
8598  8606  8613 

8472  8480  8487 
8.547  8555  8562 
8620  8627  8634 

8495 
8569 
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45 
44 
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8 

7 
7 

47 

48 
49 

8641 
8711 

8778 

8648  8655  8662 
8718  8724  8731 
8784  8791  8797 

8669  8676  8683 
8738  8745  8751 
8804  8810  8817 

8690.  8697  8704 
8758  8765  8771 
8823  8830  8836 

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42 
41 
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7 
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V.    LOaAEITHMIO   SINES  AND  COSINES. 


15 


log  sin  n° 

log 

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.4  .5  .6  j  .7   .8  .9 

1.0 

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9.8843 

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8868  8874  8880 

8887  8893  8899 

8905 

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51 
52 
53 

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8965 
9023 

8911  8917  8923 
8971  8977  8983 
9029  9035  9041 

8929  8935  8941 
8989  8995  9000 
9046  9052  9057 

8947  8953  8959 
9006  9012  9018 
9063  9069  9074 

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9023 
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38 
37 
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6 
5 
6 

54 
55 
56 

9080 
9134 
9186 

9085  9091  9096 
9139  9144  9149 
9191  9196  9201 

9101  9107  9112 
9155  9160  9165 
9206  9211  9216 

9118  9123  9128 
9170  9175  9181 
9221  9226  9231 

9134 
9186 
9236 

35 
34 
33 

6 

5 
5 

57 

58 
59 

9236 
9284 
9331 

9241  9246  9251 
9289  9294  9298 
9335  9340  9344 

9255  9260  9265 
9303  9308  9312 
9349  9353  9358 

9270  9275  9279 
9317  9322  9326 
9362  9367  9371 

9284 
9331 
9375 

32 
31 
30 
29 

5 
5 
4 

T 

60 

9.9375 

9380  93S4  9388 

9393  9397  9401 

9406  9410  9414 

9418 

61 
62 
63 

9418 
9459 
9499 

9422  9427  9431 
9463  9467  9471 
9503  9507  9510 

9435  9439  9443 
9475  9479  9483 
9514  9518  9522 

9447  9451  9455 
9487  9491  9495 
9525  9529  9533 

9459 
9499 
9537 

28 
27 
26 

4 
4 
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64 
65 
66 

9537 
9573 
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9540  9544  9548 
9576  9580  9583 
9611  9614  9617 

9551  9555  9558 
9587  9590  9594 
9621  9624  9627 

9562  9566  9569 
9597  9601  9604 
9631  9634  9637 

9573 
9607 
9640 

25 
24 
23 

4 
3 
3 

67 

68 
69 

9640 
9672 
9702 

9643  9647  9650 
9675  9678  9681 
9704  9707  9710 

9653  9656  9659 
9684  9687  9690 
9713  9716  9719 

9662  9666  9669 
9693  9696  9699 
9722  9724  9727 

9672 
9702 
9730 

22 
21 
20 
19 

3 
3 
3 
3 

70 

9.9730 

9733  9735  9738 

9741  9743  9746 

9749  9751  9754 

9757 

71 
72 
73 

9757 
9782 
9806 

9759  9762  9764 
9785  9787  9789 
9808  98119813 

9767  9770  9772 
9792  9794  9797 
9815  9817  9820 

9775  9777  9780 
9799  9801  9804 
9822  9824  9826 

9782 
9806 
9828 

18 
17 
16 

2 

2 
2 

74 
75 
76 

9828 
9849 
9869 

9831  9833  9835 
9851  9853  9855 
9871  9873  9875 

9837  9839  9841 
9857  9859  9861 
9876  9878  9880 

9843  9845  9847 
9863  9865  9867 
9882  9884  9885 

9849 
9869 
9887 

15 
14 
13 

2 

2 
2 

77 
78 
79 

9887 
9904 
9919 

9889  9891  9892 
9906  9907  9909 
9921  9922  9924 

9894  9896  9897 
9910  9912  9913 
9925  9927  9928 

9899  9901  9902 
9915  9916  9918 
9929  9931  9932 

9904 
9919 
9934 

12 
11 
10 

2 
1 
2 

80 

9.9934 

9935  9936  9937 

9939  9940  9941 

9943  9944  9945 

9946 

9 

1 

81 
82 
83 

9946 
9958 
9968 

9947  9949  9950 
9959  9960  9961 
9968  9969  9970 

9951  9952  9953 
9962  9963  9964 
9971  9972  9973 

9954  9955  9956 
9965  9966  9967 
9974  997i  9975 

9958 
9968 
9976 

8 
7 
6 

2 

1 
1 

84 
85 
86 

9976 
9983 
9989 

9977  9978  9978 
9984  9985  9985 
9990  9990  9991 

9979  9980  9981 
9986  9987  9987 
9991  9992  9992 

9981  9982  9983 
9988  9988  9989 
9993  9993  9994 

9983 
9989 
9994 

5 
4 
3 

0 
0 
0 

87 
88 
89 

9994 
9997 
9^99 

9994  9995  9995 

9998  9998  9998 

9999  0000  0000 

9996  9996  9996 
9998  9999  9999 
0000  0000  0000 

9996  9997  9997 
9999  9999  9999 
0000  0000  0000 

9997 
9999 
0000 

2 
1 

0 

0 
0 
0 

d 

1.0  1  .9   .8   .7 

.6   .5   .4 

.3  .2  .1 

.0 

u° 

pp. 

11  12  13 

14  15  16 

17  18  19 

20 

21  22  23  24  25  2627  28  29  36] 

1 
2 
3 

1  1  1 

2  2  3 

3  4  4 

1  2  2 

3  3  3 

4  5  5 

2  2  2 

3  4  4 
5  5  6 

2 
4 
6 

2  2  2 
4  4  5 
6  7  7 

2  3  3 

5  5  5 

7  8  8 

3  3  3 
5  6  6 

8  8  9 

3 
6 
9 

4 
5 
6 

4  5  5 

6  6  7 

7  7  8 

6  6  6 

7  8  8 

8  9  10 

7  7  8 
9  9  10 
10  11  11 

8 
10 
12 

8  9  9 
11  11  12 
13  13  14 

10  10  10 
12  13  13 
14  15  16 

11  1112 
14  14  15 
16  17  17 

12 
15 
18 

7 
8 
9 

8  8  9 

9  10  10 
10  11  12 

10  11  11 

11  12  13 
13  14  14 

12  13  13 

14  14  15 

15  16  17 

14 
16 
18 

15  15  16 
17  18  18 
19  20  21 

17  18  18 
19  20  21 
22  23  23 

19  20  20 
22  22  23 
24  25  26 

21 
24 

27 

LOGAKITHMIO  SINES   AND   COSINES 


lofr 

sin  )( 

log-  COS  n° 

II   ()    1   2   3 

4   5   6 

7    8   9 

10 

d 

0.0    6.  2-41';  5429  7190 

8439  9408*0200 

*0870*1450*1961 

*2419 

89.9 

0.l'7.2419  2833  3211  355S 
0.21  5429  5641  5S43  6036 
0.3l  7190  7332  7470  7604 

3880  4180  4460 
6221  6398  6568 
7734  7859  7982 

4723  4971  5206 
6732  6890  7043 
8101  8217  8329 

5429 
7190 
8439 

89.8 
89.7 
89.6 

0.4 
0.5 
O.G 

8439  8547  S651  8753  SS53  8951  9046 

9408  9494  9579  9661  9743  9822  9901 

8.0200  0272  034:  0412  0480  0548  0614 

9140  9231  9321 
9977*0053*0127 
0679  0744  0807 

9408:89.5 
*0200|  89.4 
0870  89.3 

87 
73 
63 

0.7 

0.8 
0.9 

08700931  0992  1052^1111  1169  1227 
1450  1503  1557  1609  1661  1713  1764 
1961 12009  2056  2103|2150  2196  2241 

1284  1340  1395 
1814  1863  1912 
2286  2331  2375 

1450'89.2 
1961!  89.1 
2419j89.0 

S5 
49 
44 

39 

1.0 

S.2419'2462  2505  2547 

2589  2630  2672 

2712  2753  2793 

2832  88.9 

1.1 
1.2 

lis 

2832 
3210 
3558 

2872  29112949 
3246  3282  3317 
35913624  3657 

2988  3025  3063 
3353  3388  3422 
3689  3722  3754 

3100  3137  3174 
3456  3491  3524 
3786  3817  3848 

3210  88.8 
3558:88.7 
3880|88.6 

36 
34 
32 

1.4 
1.5 
1.6 

3880 
4179 
4459 

391139413972 
4208  4237  4265 
4486  4513  4540 

4002  4032  4062 
4293  4322  4349 
4567  4593  4619 

4091  4121  4150 
4377  4405  4432 
4645  4671  4697 

4179 
4459 
4723 

88.5 
88.4 
88.3 

29 
27 
26 

1.7 
1.8 
1.9 

4723 
4971 
5206 

4748  4773  479914824  4848  4873 
4995  5019  5043^5066  5090  5113 
5228  5251  5274|5296  5318  5340 

4898  4922  4947 
5136  5160  5183 
5363  5385  5406 

4971 
5206 
5428 

88.2 
88.1 
88.0 

24 
23 
22 

2.0;S.542S 

5450  54715493^5514  5535  5557 

5578  5598  5619  5640 

87.9 

21 

2.1 
2.2 
2.3 

5640 
5842 
6035 

5661  5681  5702: 5  722  5742  5762 
5862  5881  5901  5920  5939  5959 
6054  6072  60916110  6128  6147 

5782  5802  5822 
5978  5997  6016 
6165  6183  6201 

5842 
6035 
6220 

87.8 

87.7 
87.6 

20 
19 
19 

2.4 
2.5 
2.G 

6220 
6397 
6567 

6238  6256  6274  6291  6309  6327 
6414  6431  6449  6466  6483  6500 
6584  6600  6617  6633  6650  6666 

6344  6362  6379 
6517  6534  6550 
6682  6699  6715 

6397 
6567 
6731 

87.5 
87.4 
87.3 

18 
17 
16 

2.7 

2.8 
2.9 

6731 
6889 
7041 

6747  6763  6779  6795  6810  6826 
6904  6920  693516950  6965  6981 
7056  70717086  7100  7115  7130 

6842  6858  6873 
6996  7011  7026 
7144  7159  7174 

6889 
7041 
7188 

87.2 
87.1 
87.0 

16 
15 
14 

14 

3.0 

8.7188  7202  7217  7231 

7245  7260  7274 

7288  7302  7316 

7330 

86.9 

3.1 
3.2 
3.3 

7330  7344  7358  7372 
746817482  7495  7508 
7602  7615  7628  7641 

7386  7400  7413 
7522  7535  7549 
7654  7667  7680 

7427  7441  7454 
7562  7575  7588 
7693  7705  7718 

7468 
7602 
7731 

86.8 
86.7 
86.6 

14 
14 
13 

3.4 
3.5 
3.6 

7731  7744  7756  7769  77S2  7794  7807 
7857:7869  7881  7894,7906  7918  7930 
7979|7991  8003  8015  8027  8039  8051 

7819  7832  7844 
7943  7955  7967 
8062  8074  8086 

7857 
7979 
8098 

86.5 
86.4 
86.3 

13 
12 
12 

3.7 
3.8 
3.9 

4^ 

8098:8109  8121  81338144  8156  8168 
8213  8225  8236  8248  8259  8270  8281 
8326|8337  8348  8359  S370  8381  8392 

8179  8191  8202 
8293  8304  8315 
8403  8414  8425 

8213 
8326 
8436 

86.2 
86.1 
86.0 

11 
11 
11 

11 

8.8436  8447  8457  8468  8479  8490  8500 

8511  8522  8532 

8543 

85.9 

4.1 
4.2 
4.3 

8543  8553  8564  85 75  8585  8595  8606 
8647  8658  8668  8678  8688  8699  8709 
8749  8759  8769  8780  8790  8799  8809 

8616  8627  8637 
8719  8729  8739 
8819  8829  8839 

8647 
8749 
8849 

85.8 
85.7 
85.6 

10 
10 
10 

4.4 
4.5 
4.6 

8849  8859  8869  8878  8888  8898  8908 
8946  8956  8966  8975  8985  8994  9004 
9042  9051  9060  9070  9079  90S9  9098 

8917  8927  8937 
9013  9023  9032 
9107  9116  9126 

8946 
9042 
9135 

85.5 
85.4 
85.3 

9 
10 
9 

4.7 

4.8 
4.9 

9135 
9226 
9315 

9144  9153  9162  9172  9181  9190 
^235  9244  9253  9262  9271  9280 
;324  9333  9342  9351  9359  9368 

9199  9208  9217 
9289  9298  9307 
9377  9386  9394 

9226 

9315 

9403 

0 

85.2 
85.1 
85.0 

9 
8 
9 
d 

lj 

10  1 

987|664   3   2   1 

n° 

CONTINUED. 


17 


log 

Sinn 

o 

log- 

COS  n° 

n° 

0 

12   3 

4   5   6 

7   8   9 

10 

d 

5.0 

8.9403  9412  9420  9429 

9437  9446  9455 

9463  9472  9480 

9489 

84.9  9 

5.1 
5.2 
5.3 

9489  9497  9506  9514 
9573  9581  9589  959S 
9655:9664  9672  9680 

9523  9531  9539 
9606  9614  9623 
9688  9696  9704 

9548  9556  9565 
9631  9639  9647 
9712  9720  9728 

9573 
9655 
9736 

84.8'8 
84.7:8 
84.6:8 

5.4 
5.5 
5.6 

9736  9744  9752  9760 
9816  9824  9831  9839 
9894  9901  9909  9917 

9768  9776  9784 
9S47  9855  9863 
9925  9932  9940 

9792  9800  9808 
9870  9878  9886 
9948  9955  9963 

9816  84.5'8 

9894  84.4  S 
9970  84.3,7 

5.7 

5.8 
5.9 
670 

9970 
9.0046 

0120 
9.0192 

9978  9986  9993 
0053  0061  0068 
0127  0134  0142 

*0001*0008*0016 
0075  0083  0090 
0149  0156  0163 

*0023*0031*003S 
0098  0105  0112 
0171  0178  0185 

*004684.28 
012084.118 
0192  84.0  7 

0200  0207  0214 

0221  0228  0235 

0243  0250  0257 

0264  83.9  7 

6.1 
6.2 
6.3 

0264  0271  0278  0285 
0334:0341  0348  0355 
0403  0410  0417  0424 

0292  0299  0306 
0362  0369  0376 
0431  0438  0444 

0313  0320  0327 
0383  0390  0397 
0451  0458  0465 

0334  83.87 

0403|S3.7  6 
0472;83.6  7 

6.4 
6.5 
6.6 

0472  0478  0485  0492 
0539  0545  0552  0558 
0605  0611  0618  0624 

0498  0505  0512 
0565  0572  0578 
0631  0637  0644 

0519  0525  0532 
0585  0591  0598 
0650  0657  0663 

0539J83.5  7 
060583.4:7 
0670  83.3  7 

6.7 
6.8 
6.9 
7.0 

0670  0676  0683  0689 
07340740  0746  0753 
0797  0803  0S09  0816 

0695  0702  0708 
0759  0765  0772 
0822  0828  0834 

0715  0721  0727 
0778  0784  0790 
0840  0847  0853 

0734  83.2  7 
0797  83.1  7 
0859183.0  6 

9.0859|0S65  0871  0877 

0884  0890  0896 

0902  0908  0914 

0920  82.9  6 

7.1 
7.2 
7.3 

0920  0926  0932  0938 
0981  0987  0993  0999 
1040  1046  1052  1058 

0945  0951  0957 
1005  1011  1017 
1064  1070  1076 

0963  0969  0975 
1022  1028  1034 
1081  1087  1093 

098182.8  6 
1040  82.7  6 
1099  82.6  6 

7.4 
7.5 

7.6 

1099  1105  1111  1116 
1157  1163  1168  1174 
1214  1220  1226  1231 

1122  1128  1134 
1180  1186  1191 
1237  1242  1248 

1140  1145  1151 
1197  1203  1208 
1254  1259  1265 

1157  82.5  6 
1214S2.4|6 

127182.36 

7.7 

7.8 
7.9 

8.0 

1271 

1326 

1381 

9.1436 

1276  1282  1287 
1332  1337  1343 
1387  1392  1398 

1293  1299  1304 
1348  1354  1359 
1403  1409  1414 

1310  1315  1321 
1365  1370  1376 
1419  1425  1430 

132682.2'5 
138182.115 
1436  82.0  6 

1441  1446  1452 

1457  1462  1468 

1473  1478  1484 

1489  81.9  5 

8.1 
8.2 
8.3 

1489  1494  1500  1505 
1542  1547  1553  1558 
1594  1600  1605  1610 

1510  1516  1521 
1563  1568  1574 
1615  1620  1625 

1526  1532  1537 
1579  1584  1589 
1631  1636  1641 

1542  81.8  5 
1594:81.7  5 
164681.65 

8.4 
8.5 
8.6 

1646' 1651  1656  1661 
169711702  1707  1712 
1747  1752  1757  1762 

1666  1672  1677 
1717  1722  1727 
1767  1772  1777 

1682  1687  1692 
1732  1737  1742 
1782  1787  1792 

1697 '81. 5  5 
1747  81.4  5 
1797  81.3  5 

8.7 
8.8 
8^ 

1797 
1847 

1895 

10 

1802.1807  1812 
1851  1856  1861 
1900  1905  1910 

1817  1822  1827 
1866  1871  1876 
1915  1919  1924 

1832  1837  1842 
1881  1886  1890 
1929  1934  1939 

1847  81.2  5 
1895!  81. 15 
1943  81.0  4 

9   8   7 

6   5   4 

3   2   1 

0  n° 

o 

0 

12   3 

4  5  6 

7 

8  9 

0 

S   lo^  sin 

.1 

.2 
.3 

6 

12 
IS 

6.6  7.2  7.8 
12.6  13.2  13.8 
18.6  19.2  19.8 

8.4  9  9.6 
14.4  15  15.6 
20.4  21  21.6 

10.2 
16.2 

22.2 

lO.S  11.4 
16.8  17.4 
22.8.  23.4 

0.00 

1.11 

2.40 

.7581 

.7582  8.2872 
L.7583  8.6220 

.4 
.5 
.6 

24 
30 
36 

24.6  25.2  25.8 
30.6  31.2  31.8 
36.6  37.2  37.8 

26.4  27  27.6  28.2 
32.4  33   33.6  34.2 
38.4  39  39.6  40.2 

28.8  29.4 
34.8  35.4 
40.8  41.4 

3.21 
3.85 
4.40 

1.7584  8.7482 

L.7585  8.8270 

8.8849 

.7 

.8 
.9 

42 
48 
54 

42.6  43.2  43.8 
48.6  49.2  49.8 
54.6  55.2  55.8 

14.4  45  45.6^46.2 
50.4  51  51.6,52.2 
S6.4  57  57.6I58.2 

46.8  47.4 
52.8  53.4 
58.8  59.4 

IS      VI.   LOGARITHMIO  TANGENTS  AND  COTANGENTS. 
loy  tan  H  lofiT  cotan  n° 


n^ 

.0  1  .1   .2   .3 

.4   .6   .6 

.7   .8   .9 

1.0 

d 

0 

7.'  2419  5429  7190 

8439  9409*0200 

*0870*145Q*1962 

*2419 

89 

1 
>) 

3 

S.2419  2S33  3211  3559 
5431  5643  5S45  603S 
7194  7337  74 7i  7609 

3881  4181  4461 
6223  6401  6571 
7739  7865  7988 

472i  4973  5208 
6736  6894  7046 
8107  8223  8336 

5431 
7194 
8446 

88 
87 
86 

4 
5 
(t 

S446  8554  8659  8762 

9420  9506  9591  9674 

9.0216  0289  0360  0430 

8862  8960  9056 
9756  9836  9915 
0499  0567  0633 

9150  9241  9331 
9992*0068*0143 
0699  0764  0828 

9420 

*0216 

0891 

85 
84 
83 

89 
73 
63 

7 

8 

9 

0891  0954  1015  1076 
1478  1533  1587  1640 
1997,  2046  2094  2142 

1135  1194  1252 
1693  1745  1797 
2189  2236  2282 

1310  1367  1423 
1848  1898  1948 
2328  2374  2419 

1478 
1997 
2463 

82 
81 
80 

55 
49 
44 

10 

9.2463  2507  2551  2594 

2637  2680  2722 

2764  2805  2846 

2887 

79 

41 

11 
12 
13 

2887  2927  2967  3006 
3275  3312  3349  3385 
3634  3668  3702  3736 

3046  3085  3123 
3422  3458  3493 
3770  3804  3837 

3162  3200  3237 
3529  3564  3599 
3870  3903  3935 

3275 
3634 
3968 

78 
77 
76 

38 
35 
33 

U 
15 
16 

3968  4000  4032  4064 
428r4311  4341  4371 
4575  4603  4632  4660 

4095  4127  415S 
4400  4430  4459 
4688  4716  4744 

4189  4220  4250 
4488  4517  4546 
4771  4799  4826 

4281 
4575 
4853 

75 
74 

31 
29 
27 

17 

18 
19 

20 

4853  4880  4907  4934 
511815143  5169  5195 
5370  5394  5419  5443 

4961  4987  5014 
5220  5245  5270 
5467  5491  5516 

5040  5066  5092 
5295  5320  5345 
5539  5563  5587 

5118 
5370 
5611 

72 
71 
70 

26 
25 
24 

23 

9.5611  5634  5658  5681 

5704  5727  5750 

5773  5796  5819 

5842 

69 

22 
23 

5842 
6064 
6279 

5864  5887  5909 
6086  6108  6129 
6300  6321  6341 

5932  5954  5976 
6151  6172  6194 
6362  6383  6404 

5998  6020  6042 
6215  6236  6257 
6424  6445  6465 

6064 
6279 
6486 

68 
67 
66 

22 
22 
21 

24 
25 
26 

6486 
6687 
6882 

6506  6527  6547 
6706  6726  6746 
6901  6920  6939 

6567  6587  6607 
6765  6785  6804 
6958  6977  6996 

6627  6647  6667 
6824  6843  6863 
7015  7034  7053 

6687 
6882 
7072 

65 
64 
63 

20 
19 
19 

27 

28 
29 

7072 
7257 
7438 

7090  7109  7128 
7275  7293  7311 
7455  7473  7491 

7146  7165  7183 
7330  7348  7366 
7509  7526  7544 

7202  7220  7238 
7384  7402  7420 
7562  7579  7597 

7257 
7438 
7614 

62 
61 
60 

19 
18 
17 

17 

30 

9.7614 

7632  7649  7667 

7684  7701  7719 

7736  7753  7771 

7788 

59 

31 

ti 

7788 
7958 
8125 

7805  7822  7839 
7975  7992  8008 
8142  8158  8175 

7856  7873  7890 
8025  8042  8059 
8191  8208  8224 

7907  7924  7941 
8075  8092  8109 
8241  8257  8274 

7958 
8125 
8290 

58 
57 
56 

17 
16 
16 

It 

36 

8290 
8452 
8613 

8306  8323  8339 
8468  8484  8501 
8629  8644  8660 

8355  8371  8388 
8517  8533  8549 
8676  8692  8708 

8404  8420  8436 
8565  8581  8597 
8724  8740  8755 

8452 
8613 
8771 

55 
54 
53 

16 
16 
16 

37 
38 
39 

40 

8771  8787  SS03  8818 
89281  8944  8959  S97i 
9084  9099  9115  9130 

8S34  SS50  8865 
8990  9006  9022 
9146  9161  9176 

8881  8897  8912 
9037  9053  9068 
9192  9207  9223 

8928 
9084 
9238 

52 
51 
50 

16 
16 
15 

16 

9.9238  9254  9269  9284 

9300  9315  9330 

9346  9361  9376 

9392 

49 

41 
42 
43 

9392'  9407  9422  9438 
9544  9560  9575  9590 
9697  9712  9727  9742 

9453  9468  9483 
9605  9621  9636 
9757  9772  9788 

9499  9514  9529 
9651  9666  9681 
9803  9818  9833 

9544 
9697 

9848 

48 
47 
46 

15 
16 
15 

44 
45 
46 

9848  9864  9879  9S94 

0.0000  0015  0030  0045 

0152,0167  01S2  0197 

9909  9924  9939 
0061  0076  0091 
0212  0228  0243 

9955  9970  9985 
0106  0121  0136 
0258  0273  0288 

*0000 
0152 
0303 

45 
44 
43 

15 
16 
15 

47 

48 
49 

0303  0319  0334  03491  0364  0379  0395 
0456  0471  0486  05011  0517  0532  0547 
0608  0624  0639  0654  0670  0685  0700 

0410  0425  0440 
0562  0578  0593 
0716  0731  0746 

0456 
0608 
0762 

42 
41 
40 

16 
15 
16 

1.0   .9   .8   .7   .6   .5   .4 

.3   .2   .1 

.0  In^ 

VI.  LOGARITHMIC  TANGENTS  AND  COTANGENTS.      19 
logtan?t° log  cotan  n° 


.0 


0.0762 
0916 
10 

1229 
1387 
1548 
1710 
1875 
2042 
2212 


0.2386 
2562 
2743 
2928 
3118 
3313 
3514 
3721 
3936 
4158 


0.4389 
4630 
4882 
5147 
5425 
5719 
6032 
6366 
6725 
7113 


.2   .3 


0777 
0932 
1088 
1245 
1403 
1564 
1726 
1891 
2059 
2229 


0793  0808 
0947  0963 
1103  1119 
1260  1276 
1419  1435 
1580  1596 
1743  1759 
1908  1925 
2076  2093 
2247  2264 


2403 
2580 
2762 
2947 
3137 
3333 
3535 
3743 
3958 
4181 


2421  2438 
2598  2616 
2780  2798 
2966  2985 
3157  3176 
33S3  3313 
3555  3576 
3764  3785 
3980  4002 
4204  4227 


0.7537 
8003 
8522 
9109 
9784 

1.0580 
1554 
2806 
4569 
7581 


1.0 


4413 
4655 
4908 
5174 
5454 
5750 
6065 
6401 
6763 
7154 


4437  4461 
4680  4705 
4934  4960 
5201  5229 
5483  5512 
5780  5811 
6097  6130 
6436  6471 
6800  6838 
7195  7236 


.5 


0824 
0978 
1135 
1292 
1451 
1612 
1776 


0839  0854 
0994  1010 
1150  1166 
1308  1324 
1467  1483 
1629  1645 
1792  1809 


1941  1958  1975 
2110  2127  2144 
2281  2299  2316 


2456  2474  2491 


2634 
2817 
3004 
3196 
3393 
3596 
3806 
4024 
4250 


2652  2670 
2835  2854 
3023  3042 
3215  3235 
3413  3433 
3617  3638 
3828  3849 
4046  4068 
4273  4296 


7581 
8052 
8577 
9172 
9857 
0669 
1664 
2954 
4792 
8038 


7626  7672 
8102  8152 
8633  8690 
9236  9301 
9932*0008 
0759  0850 
1777  1893 
3106  3264 
5027  5275 
8550  9130 


4484 
4730 
4986 
5256 
5541 
5842 
6163 


4509  4533 
4755  4780; 
5013  5039 
5284  5312 
5570  5600 
5873  5905 
6196  6230 


.9 


0870  0885  0901 
1025  1041  1056 
1182  1197  1213 
1340  1356  1371 
1499  1516  1532 
1661  1677  1694 
1825  1842  1858 


1992  2008 
2161  2178 
2333  2351 


2025 
2195 
2368 


2509  2527 
2689  2707 
2872  2891 
3061  3080 
325 t  3274 
3453  3473 
3659  3679 
3871  3892 
4091  4113 
4319  4342 


2545 
2725 
2910 
3099 
3294 
3494 
3700 
39^4 
4136 
4366 


6507  6542  6578 
6877  6915  6954 
7278  7320  7363 


7718  7764  7811 


8203 
8748 
9367 
^0085^ 
0944 
2012 
3429 
5539 
9800^ 


8255  8307 
8806  8865 
9433  9501 
0164*0244 
1040  1138 
2135  2261 
3599  3777 
5819  6119 
0591*1561 


4557  4581 
4805  4831 
5066  5093 
5340  5368 
5629  5659 
5936  5968 
6264  6298 


4606 
4857 
5120 
5397 
5689 
6000 
6332 


6615  6651  6688 
6994  7033  7073 
7406  7449  7493 


.6 


A 


7858  7906  7954 
8360  8413  8467 
8924  8985  9046 
9570  9640  9711 
"0326*0-109*0494 
1238  1341  1446 
2391  2525  2663 
3962  4155  4357 
6441  6789  7167 
■2810*4571*7581 


1.0 


091639 

107238 

122937 

138730 

1548  35 

171034 

187533 

2042  32 

2212!31 

2386'30 

2562  29 

274328 

2928  27 

3118  26 

3313  25 

3514  24 

3721  23 

3936 '22 

415821 

438920 

4630  19 

4882 

5147 

5425 

5719 

6032^14 

63o6[13 

6725  12 

711311 


7537  10:44 


8003 
8522 
9109 
9784 
►05  SO 
1554 
2806 
4569 
7581 


.3 


.1  I  .0 


9j49 

855 
7  63 


pp. 


31  32  33 


34  35  36 


17  38  39 


40 

41  42  43  44  45  46 

47  48  49 

4 
8 
12 

J    4  4 
'%    8  9 
12  13  13 

4  5  5 
9  9  9 
13  14  14 

5  5  5 
9  10  10 
14  14  15 

16 

20 
24 

16  17  17 
21  21  22 
25  25  26 

18  18  18 
22  23  23 

26  27  28 

19  19  20 

24  24  25 
28  29  29 

28 
32 
36 

29  29  30 
33   34  34 
37  38  39 

3132  32 
35  36  37 
40  41  41 

33  34  34 
38  38  39 
42  43  44 

50 


3  3  3 
6  6  7 
9  10  10 
12  13  13 
16  16  17 
19  19  20 
22  22  23 
25  26  26 
28  29  30 


3  4  4 
7  7  7 
10  11  11 
14  14  14 
17  18  18 
20  21  22 
24  25  25 
27  28  29 
31  32  32 


4  4  4 
7  8  8 
11  11  12 
15  15  16 
19  19  20 
22  23  23 
26  27  27 
30  30  31 
33  34  35 


20      VI.   LOGAEITHMIO  TANGENTS  AND  COTANGENTS 

loj7  tan  «  '  1<>^  votannP 


CONTINUED. 


21 


log  tan  n° 

loff  cotan 

n° 

n° 

0 

1   2   3 

4   5   6 

7   8   9 

10 

84.9 

d 

5.0 

8.942( 

)  9428  9437  9446 

9454  9463  9472 

9480  9489  9497 

9506 

5.1 
5.2 
5.3 

950( 
959] 
967^ 

)  9515  9523  9532 
9599  9608  9616 
9682  9690  9699 

9540  9549  9557 
9624  9633  9641 
9707  9715  9723 

9565  9574  9582 
9649  9657  9666 
9731  9739  9747 

9591 
9674 
9756 

84.8 
84.7 
84.6 

9 
8 
9 

5.4 
5.5 
5.6 

975^ 
9S36 
991i 

9764  9772  9780 
9844  9852  9860 
9922  9930  9938 

9788  9796  9S04 
9867  9875  9883 
9946  9953  9961 

9812  9820  9828 
9891  9899  9907 
9969  9977  9984 

9836 
9915 
9992 

84.5 
84.4 
84.3 

8 
8 
8 

5.7 

5.8 
5.9 

9992 

9.006e 

0142 

*0000*0007*0015 
0075  0083  0090 
0150  0157  0165 

*0022*0030*003S 
0098  0105  0113 
0172  0180  0187 

*0045*0053*0060 
0120  0128  0135 
0194  0202  0209 

^0068 
0143 
0216 
0289 

84.2 
84.1 
84.0 

83.9 

8 
8 
7 

8 

6.0 

9.021^ 

0223  0231  0238 

0245  0253  0260 

0267  0274  0281 

6.1 
6.2 
6.3 

02SS 
036C 
043C 

0296  0303  0310 
0367  0374  0381 
0437  0444  0451 

0317  0324  0331 
0388  0395  0402 
0457  0464  0471 

0338  0346  0353 
0409  0416  0423 
0478  0485  0492 

0360 
-0430 
0499 

83.8 
83.7 
83.6 

7 
7 
7 

6.4 
6.5 
6.6 

049? 
0567 
0633 

0506  0512  0519 
0573  0580  0587 
0640  0647  0653 

0526  0533  0540 
0593  0600  0607 
0660  0667  0673 

0546  0553  0560 
0614  0620  0627 
0680  0686  0693 

0567 
0633 
0699 

83.5 
83.4 
83.3 

7 
6 
6 

6.7 
6.8 
6.9 

069C 
07(A 
082S 

0/06  0712  0719 
0771  0777  0784 
0835  0841  0847 

0725  0732  0738 
0790  0796  0803 
0854  0860  0866 

0745  0751  0758 
0809  0816  0822 
0873  0879  0885 

0764 
0828 
0891 

83.2 
83.1 
83.0 

82.9 

6 
6 
6 

7 

7.0 

9.0891 

0898  0904  0910 

0916  0923  0929 

0935  0941  0947 

0954 

7.1 
7.2 
7.3 

095^ 
101  = 
107^ 

0960  0966  0972 
1021  1027  1033 
1082  1088  1094 

0978  0984  0991 
1039  1045  1051 
1100  1106  1112 

0997  1003  1009!  1015 
1058  1064  1070  1076 
1117  1123  1129  1135 

82.8 
82.7 
82.6 

6 
6 
6 

7.4 
7.5 
7.6 

113f 
119^ 
1252 

1141  1147  1153 
1200  1206  1212 
1258  1264  1270 

1159  1165  1171 
1218  1223  1229 
1276  1281  1287 

1177  1183  1188 
1235  1241  1247 
1293  1299  1304 

1194 
1252 
1310 

82.5 
82.4 
82.3 

6 
5 
6 

7.7 

7.8 
7.9 

131C 
136/ 
142^ 

)  1316  1321  1327 
1372  1378  1384 
1428  1434  1439 

1333  1338  1344 
1389  1395  1400 
1445  1450  1456 

1350  1355  1361 
1406  1412  1417 
1461  1467  1473 

1  1367 
1423 
1478 

82.2 
82.1 

82.0 

6 
6 

5 

8.0 

9.1475 

1484  1489  1494 

1500  1503  151] 

1516  1522  1527 

1533 

81.9 

6 

8.1 

8.2 
8.3 

1533 
158/ 
164C 

1538  1544  1549 

1592  1597  1603 

)  16^5  1651  1656 

1554  1560  1565 
1608  1613  1619 
1661  1667  1672 

1571  1576  1581 
1624  1629  1635 
1677  1682  1688 

1587 
1640 
1693 

81.8 
81.7 
81.6 

6 

5 
5 

8.4 
8.5 
8.6 

16931  1698  1703  1709 
17i5  1750  1755  1761 
1797  1802  1807  1812 

1714  1719  1724 
1766  1771  1776 
1817  1822  1827 

1729  1735  1740 
1781  1786  1791 
1832  1837  1842 

1745 
1797 
1848 

81.5 
81.4 
81.3 

5 
6 
6 

8.7 
8.8 
8.9 

1848  1853  1858  1863 
1898  1903  1908  1913 
1948'  1953  1958  1963 

1868  1873  1878;  1883  1888  1893 
1918  1923  1928  1933  1938  1943 
1968  1973  1977  1982  1987  1992 

1898 
19-18 
1997 

81.2 
81.1 
81.0 

11° 

5 
5 

5 

10  !   9   8   7 

6   5   4    3   2   1  1  0 

f 

0 

1 

0 

0 
.16 

1   2   3 

4   5   6 

7   8   9 

o 

,T  log  tan 

.016  .03  .05 
.183  .2  .216 

.06  .083  .1 
.23  .25  .26 

.116  .13  .15 
.283  .3  .316 

0.00 
1.28 
1.98 

1.7581 

1.7580  8.3492 
1.7579  8.5387 

2 
8 

.3 
.5 

.35  .36  .383 
.516  .53  .55 

.^    »416  .43 
.56  .583  .6 

.45  .46  .483 
.616  .63  .65 

2.49 
2.91 
3.27 

1.7578  8.6384 
1.7577  8.7061 
1.7576  8.7569 

4 
5 

.6 

.83 

.683  .7  .716 
.85  .86  .883 

.73  .75  .76 
.9  .916  .93 

.783  .8  .816 
.95  .96  .983 

3.60 
3.90 
4.18 

1.7575  8.7988 

1.7574  8.8336 

8.8638 

22  VII.   LOGAKITHMIO   SINES   AND  COSINES 


u"^^ 

0' 

10' 

20' 

30' 

40'  50'  j  60' 

d 

PP 

0 

7.4637 

7648 

9408 

*065S*1627*2419i89 

45  46  47  48  49 

1 
2 

3 

S.2419 
542b 
7ISS 

3088 
5776 
7423 

3668 
6097 
7645 

4179 
6397 

7857 

4637  5050 
6677  6940 
8059  8251 

5428  88 
7188  87 
8436  86 

5  5  5  5  5 
9  9  9  10  10 
14  14  14  14  15 

4 
o 
0 

8436 

9403 

9.0192 

8613 
9545 
0311 

8783 
9682 
0426 

8946 
9816 
0539 

9104  9256 
9945*0070 
0648  0755 

9403 

*0192 

0859 

85 
84 
83 

18  18  19  19  20 
23  23  24  24  25 
27  28  28  29  29 

7 

8 
9 

0S.S9 
1436 
1943 

0961 
1525 
2022 

1060 
1612 
2100 

1157 
1697 
2176 

1252  1345 
1781  1863 
2251  2324 

1436 
1943 
2397 

82 
81 
80 

91 
80 
73 

32  32  33  34  34 
36  37  38  38  39 
41  41  42  43  44 

9.2397 

2468 

2538 

2606 

2674  2740 

2806 

79 

66 

40  41  42  43  44 

11 
12 
13 

2806 
3179 
3521 

2870 
3238 
3575 

2934 
3296 
3629 

2997 
3353 
3682 

3058  3119 
3410  3466 
3734  3786 

3179 
3521 
3837 

78 
77 
76 

60 
55 
51 

4  4  4  4  4 
8  8  8  9  9 
12  12  13  13  13 

U 
15 
10 

3837 
4130 
4403 

3887 
4177 
4447 

3937 
4223 
4491 

3986 
4269 
4533 

4035  4083 
4314  4359 
4576  4618 

4130 
4403 
4659 

75 
74 
73 

47 
44 
41 

16  16  17  17  18 
20  21  21  22  22 
24  25  25  26  26 

17 

18 
19 

20 

4659 
4900 
5126 

4700 
4939 
5163 

4741 
4977 
5199 

4781 
.5015 
5235 

4821  4861 
5052  5090 
5270  5306 

4900 
5126 
5341 

72 
71 
70 

39 
36 
35 

28  29  29  30  31 
32  33   34  34  35 
36  37  38  39  40 

9.5341 

5375 

5409 

5443 

5477  5510 

5543 

69 

33 

35  36  37  38  39 

21 
^2 
23 

5543 
5736 
5919 

5576 
5767 
5948 

5609 
5798 
5978 

5641 

5828 
6007 

5673  5704 
5859  5889 
6036  6065 

5736 
5919 
6093 

68 
67 
66 

32 
30 
28 

4  4  4  4  4 

7  7  7  8  8 

11  11  11  11  12 

24 
25 
26 

6093 
6259 
6418 

6121 
6286 
6444 

6149 
6313 
6470 

6177 
6340 
6495 

6205  6232 
6366  6392 
6521  6546 

6259 
6418 
6570 

65 
64 
63 

27 
26 
24 

14  14  15  15  16 

15  18  19  19  20 
21  22  22  23  23 

27 

28 
29 

6570 
6716 
6856 

6595 
6740 
6878 

6620 
6763 
6901 

6644 
6787 
6923 

6668  6692 
6810  6833 
6946  6968 

6716 
6856 
6990 

62 
61 
60 

24 
23 
22 

25  25  26  27  27 
28  29  30  30  31 
32  32  33   34  35 

30 

9.6990 

7012 

7033 

7055 

7076'  7097 

7118 

59 

21 

30  31  32  33  34 

31 
32 
33 

7118 
7242 
7361 

7139 
7262 
7380 

7160 
7282 
7400 

7181 
7302 
7419 

7201  7222 
7322  7342 
7438  7457 

7242 
7361 
7476 

58 
57 
56 

20 
19 
19 

3  3  3  3  3 
6  6  6  7  7 
9  9  10  10  10 

34 
35 
36 

7476 
7586 
7692 

7494 
7604 
771$ 

7513 
7622 
7727 

7531 
7640 
7744 

7550  7568 
7657  7675 
7761  7778 

7586 
7692 
7795 

55 
54 
53 

18 
17 
17 

12  12  13  13  14 
15  16  16  17  17 
18  19  19  20  20 

37 
38 
39 

40 

7795 
7893 
7989 

7811 
7910 
8004 

7828 
7926 
8020 

7844 
7941 
8035 

7861  7877 
7957  7973 
8050  8066 

7893 
7989 
8081 

52 
51 
50 

16 
16 
15 

21  22  22  23  24 
24  25  26  26  27 
27  28  29  30  31 

9.8081 

8096 

8111 

8125 

8140  8155 

8169 

49 

14 

25  26  27  28  29 

41 
42 
43 

8169 

8255 
8338 

8184 
8269 
8351 

8198 
8283 
8365 

8213 
8297 
8378 

8227  8241 
8311  8324 
8391  8405 

8255*48 
8338  47 
8418  46 

14 
14 
13 

3     3     3     3     3 

5  5  5  6  6 
8  8  8  8  9 

44 
45 
46 

8418 
8495 
8569 

8431 
8507 
8582 

8444 
8520 
8594 

8457 
8532 
8606 

8469  8482 
8545  8557 
8618  8629 

8495 
8569 
8641 

45 
44 
43 

13 

12 
12 

10  10  11  11  12 
13  13  14  14  15 
15  16  16  17  17 

47 

48 
49 

8641 
8711 
8778 

8653 
8722 
8789 

50' 

8665 
8733 
8800 

8676 
8745 
8810 

86SS  8699 
8756  8767 
SS21  8832 

8711 
8778 
8843 

42 
41 
40 

12 
11 
11 

18  IS  19  20  20 
20  21  22  22  23 
23  23  24  25  26 

60' 

40' 

30' 

20'  10' 1  0' 

11° 

d 

rOK  MINUTES. 


23 


log: 

sin 

logr 

cos 

n° 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

(1 

PP       1 

50 

9.8S43 

8853 

8864 

8874 

8884  8895 

8905 

39 

10 

20 

21 

22 

23  24 

51 
52 
53 

8905 
8965 
9023 

8915 
8975 
9033 

8925 
8985 
9042 

8935 
8995 
9052 

8945  8955 
9004  90i4 
9061  9070 

8965 
9023 
9080 

38 
37 
36 

10 
9 
10 

2 
4 
6 

2 
4 
6 

2 

4 

7 

2  2 

5  5 

7  7 

54 
55 
56 

9080 
9134 
9186 

9089 
9142 
9194 

9098 
9151 
9203 

9107 
9160 
9211 

9116  9125 
9169  9177 
9219  9228 

9134 
9186 
9236 

35 
34 
33 

9 
9 

8 

8 
10 
12 

8 
11 
13 

9 
11 
13 

9  10 
12  12 
14  14 

57 

58 
59 

9236 
9284 
9331 

9244 
9292 
9338 

9252 
9300 
9346 

9260 
9308 
9353 

9268  9276 
9315  9323 
9361  9368 

9284 
9331 
9375 

32 
31 
30 

8 
8 
7 

14  15  15  16  17 
16  17  18  18  19 

15  19  20  21  22 

60 

9.9375 

9383 

9390 

9397 

9404  9411 

9418 

29 

7 

15 

16  17  18  19| 

61 
62 
63 

9418 
9459 
9499 

9425 
9466 
9505 

9432 
9473 
9512 

9439 
9479 
9518 

9446  9453 
9486  9492 
9524  9530 

9459 
9499 
9537 

28 
27 
26 

6 

7 

7 

2 
3 

5 

2 
3 
5 

2 
3 

5 

2  2 

4  4 

5  6 

64 
65 
66 

9537 
9573 
9607 

9543 
9579 
9613 

9549 
9584 
9618 

9555 
9590 
9624 

9561  9567 
9596  9602 
9629  9635 

9573 
9607 
9640 

25 
24 
23 

6 

5 
5 

6 
8 
9 

6 
8 
10 

7 
9 
10 

7  8 
9  10 
11  11 

67 

68 
69 

9640 
9672 
9702 

9646 
9677 
9706 

9651 
9682 
9711 

9656 
9687 
9716 

9661  9667 
9692  9697 
9721  9725 

9672 
9702 
9730 

22 
21 
20 

5 

5 
5 

11 

12 

11 
13 
14 

12 
14 
15 

13  13 

14  15 
16  17 

70 

9.9730 

9734 

9739 

9743 

9748  9752 

9757 

19 

5 

10  11 

12 

13  14 

71 

72 
73 

9757 
9782 
9806 

9761 
9786 
9810 

9765 
9790 
9814 

9770 
9794 
9817 

9774  9778 
979S  9802 
9821  9825 

9782 
9806 
9828 

18 
17 
16 

4 
4 
3 

1 
2 
3 

1 
2 
3 

1 
2 
4 

1  1 

3  3 

4  4 

74 
75 
76 

9828 
9849 
9869 

9832 
9853 
9872 

9836 
9856 
9875 

9839 
9859 

9878 

9843  9846 
9863  9866 
9881  9884 

9849 
9869 
9887 

15 
14 
13 

I 

3 

4 

5 
6 

4 
6 

7 

5 
6 

7 

5  6 

7  7 

8  8 

77 
78 
79 

9887 
9904 
9919 

9890 
9907 
9922 

9893 
9909 
9924 

9896 
9912 
9927 

9899  9901 
9914  9917 
9929  9931 

9904 
9919 
9934 

12 
11 
10 

3 
2 
3 

7 
8 
9 

8  8 

9  10 
10  11 

9  10 
10  11 
12  13 

80 

9.9934 

9936 

9938 

9940 

9942  9944 

9946 

9 

2 

5 

6 

7 

8  9 

81 
82 
83 

9946 
9958 
9968 

9948 
9959 
9969 

9950 
9961 
9971 

9952 
9963 
9972 

9954  9956 
9964  9966 
9973  9975 

9958 
9968 
9976 

8 
7 
6 

2 

2 
1 

1 
1 
2 

1 
1 
2 

1 
1 
2 

1  1 

2  2 
2  3 

84 
85 
86 

9976 
9983 
9989 

9977 
9985 
9990 

9979 
9986 
9991 

9980 
9987 
9992 

9981  9982 
9988  9989 
9993  9993 

9983 
9989 
9994 

5 
4 
3 

1 
0 

1 

2 
3 
3 

2 
3 
4 

3 

4 
4 

3  4 

4  5 

5  5 

87 
88 
89 

9994 
9997 
9999 

9995 
9998 
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9995 

9998 

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9996 
9999 
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9996  9997 
9999  9999 
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9997 
9999 
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2 

1 
0 

0 
0 
0 

4 
4 
5 

4 

5 
5 

5 
6 
6 

6  6 

6  7 

7  8 

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50' 

40' 

30' 

20'  10' 

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1 
0 
0 
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1 
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2 

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3  4 

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1  2 

2  2 
2  2 

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S   1 

og 

sin 

0  00 

1  51 

2  50 

1 
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0  3.5363 

1  3.5364  8.5090 
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5  07 
5  32 
5  56 

30 
33 
35 

7  3.5369 
I    3.5370 
5  3.5371 

8.95 
8.95 
9.01 

03 
42 
44 

8  32 

4  08 
4  39 

2 
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2 

2  3.5366  8.7898 
\S    3.5367  8.8578 
■9  3.5368  8.9089 

6  18 
6  39 
6  59 

37 
39 
41 

S  3.5372 
9  3.5373 
9 

9.0-1 
9.0^ 
9.05 

03 
37 
49 

1 

1 
1 

2 

2 

2  3 

2  3 

3  4 

2J     VIII. 

lot;  tllll 


LOGARITHMIC  TANGENTS  AND  COTANGENTS 
Utii  cotan 


n^" 

0' 

10' 

20' 

30' 

40' 

50'  j  60' 

d 

PP 

0 

7.4637 

7648 

940"; 

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89 

95  96  97  98  99 

8.2419 
5431 
7194 

3089 
5779 
7429 

3669 
6101 
7652 

4181 
6401 

786i 

4638 
6682 
8067 

5053 
6945 
8261 

5431 
7194 
8446 

88 
87 
86 

10  10  10  10  10 
19  19  J9  20  20 
29  29  29  29  30 

4 
5 

« 

8446 

9420 

9.0216 

8624 
9563 
0336 

8795 
9701 
0453 

8960 
9836 
0567 

9118 
9966 
0678 

9272 
*0093 
0786 

9420|  85 
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0891,83 

105 

38  38  39  39  40 
48  48  49  49  50 

57  58  58  59  59 

7  i  0891 

8  1478 

9  1997 

099i 
1569 
2078 

1096 
1658 
2158 

1194 
1745 
2236 

1291 
1831 
2313 

1385 
1915 
2389 

1478 
1997 
2463 

80 

93 

82 
74 

67  67  68  69  69 
76  77  78  78  79 
86  86  87  88  89 

10  9.2463 

2536 

2609 

2680 

2750 

2819 

2887 

79 

68 

90  91  92  93  94 

11  2887 

12  3275 

13  3634 

2953 
3336 
3691 

3020  3085 
3397  3458 
3748  3804 

3149 
3517 
3859 

3212 
3576 
3914 

3275 
3634 
3968 

78 
77 
76 

63 

58 
54 

9  9  9  9  9 
18  IS  18  19  19 

27  27  28  28  28 

14  3968 

15  42SI 
1<>   457i 

4021 
4331 
4622 

4074 
4381 
4669 

4127 
4430 
4716 

4178 
4479 
4762 

4230 
4527 
4808 

4281 
4575 
4853 

75 
74 
73 

51 
48 
45 

36  36  37  37  38 
45  46  46  47  47 
54  55  55  56  56 

17 
IS 

ly 

4853 
5118 
5370 

4898 
5161 
5411 

4943 
5203 
5451 

4987 
5245 
5491 

5031 
5287 
5531 

5075 
5329 
5571 

5118 
5370 
5611 

72 
71 
70 

43 
41 
40 

63  64  64  65  66 
72  73  74  74  75 
81  82  83  84  85 

20 

9.5611 

5650 

5689 

5727 

■5766 

5804 

5842 

69 

38 

85  86  87  88  89 

21 
22 
23 

5842 
6064 
6279 

5879 
6100 
6314 

5917 
6136 
6348 

5954 
6172 
6383 

5991 
6208 
6417 

6028 
6243 
6452 

6064 
6279 
6486 

68 
67 
66 

36 
36 
34 

9  9  9  9  9 

17  17  17  18  18 
26  26  26  26  27 

24 
25 
26 

6486 
6687 
6882 

6520 
6720 
6914 

6553 
6752 
6946 

6587 
6785 
6977 

6620 
6817 
7009 

6654 
6850 
7040 

6687 
6882 
7072 

65 
64 
63 

33 
32 
32 

34  34  35  35  36 
43  43  44  44  45 
51  52  52  53  53 

27 

28 
29 

7072 
7257 
7438 

7103 
7287 
7467 

7134 
7317 
7497 

7165 
7348 
7526 

7196 
7378 
7556 

7226 
7408 
7585 

7257 
7438 
7614 

62 
61 
60 

31 
30 
29 

60  60  61  62  62 
68  69  70  70  71 

77  77  78  79  SO 

30  ^9.7614 

7644 

7673 

7701 

7730 

7759 

7788 

59 

29 

80  81  82  83  84 

31 
32 
33 

7788 
7958 
8125 

7816 
7986 
8153 

7845 
8014 
8180 

7873 
8042 
8208 

7902 
8070 
8235 

7930 
8097 
8263 

7958 
8125 
8290 

58 
57 
56 

28 
28 
27 

S  8  S  8  8 
16  16  16  17  17 
24  24  25  25  25 

34 
35 

36 

8290 
8452 
8613 

8317 
8479 
8639 

8344 
8506 
8666 

8371 
8533 
8692 

8398 
8559 
8718 

8425 
8586 

8745 

8452 
8613 
8771 

55 
54 
53 

27 
27 
26 

32  32  33  33   34 
40  41  41  42  42 
48  49  49  50  50 

37 
38 
39 

8771 
8928 
9084 

8797 
8954 
9110 

SS24 
8980 
9135 

8850 
9006 
9161 

8876 
9032 
9187 

8902 
9058 
9212 

S92S 
9084 
9238 

52 
51 
50 

26 
26 
26 

56  57  57  58  59 
64  65  66  66  67 
72  73  74  75  76 

40 

9.9238 

9264 

9289 

9315 

9341 

9366 

9392 

49 

26 

75  76  77  78  79 

41  9392 

42  9544 

43  9697 

9417 
9570 
9722 

9443 
9595 
9747 

9468 
9621 
9772 

9494 
9646 
9798 

9519 
9671 
9823 

9544 
9697 
9848 

48 
47 
46 

25 
26 

25 

8  8  8  8  8 
15  15  15  16  16 
23  23  23  23  24 

44  9848 

45  0.0000 

46  0)52 

9874 
0025 
0177 

9899 
0051 
0202 

9924 
0076 
0228 

9949 
0101 
0253 

9975 
0126 
0278 

*0000 
0152 
0303 

45 
44 
43 

25 
26 

25 

30  30  31  31  32 
38  38  39  39  40 
45  46  46  47  47 

47  0303 

48  04 .56 

49  '  0608 

0329 
0481 
0634 

0354 
0506 
0659 

0379 
0532 
0685 

0405 
0557 
0711 

0430 
0583 
0736 

0456 
0608 
0762 

42 
41 
40 

26 
25 
26 

53  53  54  55  55 
60  61  62  62  63 
68  68  69  70  71 

60' 

50' 

40' 

30' 

20' 

10' j  0' 

n^ 

d 

FOE  MINUTES. 

25 

log-  tan                                                     log-  cotan 

n° 

0' 

10'     20'    30'   40'    50' 

60' 

d 

PP 

50 

0.0762 

0788  0813  0839  0865  0890 

0916 

39 

26 

70  71  72  73  74 

51 
52 
53 

0916 
1072 
1229 

0942  0968  0994  1020  1046 
1098  1124  1150  1176  1203 
1255  1282  1308  1334  1361 

1072 
1229 
1387 

38 
37 
36 

26 
26 
26 

7  7  7  7  7 
14  14  14  IS  15 
21  21  22  22  22 

54 
55 
56 

1387 
1548 
1710 

1414  1441  1467  1494  1521 
1575  1602  1629  1656  1683 
1737  1765  1792  1820  1847 

1548 
1710 

1875 

35 
34 
33 

27 
27 
28 

28  28  29  29  30 
35  36  36  37  37 
42  43  43  44  44 

57 

58 
59 

1875 
2042 
2212 

1903  1930  1958  1986  2014 
2070  2098  2127  2155  2184 
2241  2270  2299  2327  2356 

2042 
2212 
2386 

32 
31 
30 

28 
28 
30 

49  50  50  51  52 
56  57  58  58  59 
63  64  65  66  67 

60 

0.2386 

2415  2444  2474  2503  2533 

2562 

29 

29 

65  66  67  68  69 

61 
62 
63 

2562 
2743 
2928 

2592  2622  2652  2683  2713 
2774  2804  2835  2866  2897 
2960  2991  3023  3054  30S6 

2743 
2928 
3118 

28 
27 
26 

30 
31 
32 

7  7  7  7  7 
13  13  13  14  14 
20  20  20  20  21 

64 
65 
66 

3118 
3313 
3514 

3150  3183  3215  3248  3280 
3346  3380  3413  3447  3480 
3548  3583  3617  3652  3686 

3313 
3514 
3721 

25 
24 
23 

33 

34 
35 

26  26  27  27  28 
33  33  34  34  35 
39  40  40  41  41 

67 

68 
69 

3721 
3936 
4158 

3757  3792  3828  3864  3900 
3972  4009  4046  4083  4121 
4196  4234  4273  4311  4350 

3936 

4158 
4389 

21 
20 

36 
37 
39 

46  46  47  48  48 
52  53  54  54  55 
59  59  60  61  62 

70 

0.4389 

4429  4469  4509  4549  4589 

4630 

19 

41 

60  61  62  63  64 

71 
72 
73 

4630 
4882 
5147 

4671  4713  4755  4797  4839 
4925  4969  5013  5057  5102 
5192  5238  5284  5331  5378 

4882 
5147 
5425 

18 
17 
16 

43 
45 
47 

6  6  6  6  6 
12  12  12  13  13 
18  18  19  19  19 

74 
75 
76 

5425 
5719 
6032 

5473  5521  5570  5619  5669 
5770  5822  5873  5926  5979 
6086  6141  6196  6252  6309 

5719 
6032 
6366 

15 
14 
13 

50 
53 

57 

24  24  25  25  26 
30  31  31  32  32 
36  37  37  38  38 

77 
78 
79 

6366 
6725 
7113 

6424  6483  6542  6603  6664 
6788  6851  6915  6980  7047 
7181  7250  7320  7391  7464 

6725 
7113 
7537 

12 
11 
10 

61 
66 
73 

42  43  43  44  45 
48  49  50  SO  51 
54  55  56  57  58 

80 

0.7537 

7611  7687  7764  7842  7922 

8003 

9 

81 

55  56  57  58  59 

81 

82 
83 

8003 
8522 
9109 

8085  8169  8255  8342  8431 
8615  8709  8806  8904  9005 
9214  9322  9433  9547  9664 

8522 
9109 
9784 

8 
7 
6 

91 
104 
120 

6  6  6  6  6 
11  11  11  12  12 
17  17  17  17  18 

84 
85 
86 

9784 

1.0580 

1554 

9907*0034*0164*0299*0437 
0728  0882  1040  1205  1376 
1739  1933  2135  2348  2571 

*0580 
1554 
2806 

5 
4 
3 

22  22  23  23  24 
28  28  29  29  30 
33  34  34  35  35 

87 
88 
89 

2806 
4569 
7581 

3055  3318  3599  3899  4221 
4947  5362  5819  6331  6911 
8373  9342*0591*2352*5363 

4569 
7581 

2 

1 
0 

39  39  40  41  41 
44  45  46  46  47 
50  50  51  52  53 

60' 

50'     40'   30'   20'    10' 

0' 

n° 

d 

50  51  52  53  54 

5  5  5  5  5 
10  10  10  11  11 
15  15  16  16  16 

0     r 

'        T      log  tan 

o     r 

'        T       log  tan 

0  00 

0  44 

1  41 

0    3.5363 
44    3.5362    8.1072 
101    3.5361    8.4682 

4  19 
4  34 
4  49 

259    3.5354    8.8778 
274    3.5353    8.9024 
289    3.5352    8.9256 

2  15 

2  43 

3  06 

135    3.5360    8.5943 
163    3.5359    8.6762 
186    3.5358    8.7337 

5  03 
5  16 

5  28 

303    3.5351    S.9463 
316    3.5350    8.9646 
328    3.5349    8.9809 

20  20  21  21  22 
25  26  26  27  27 
30  31  31  32  32 

3  27 

3  46 

4  03 

207    3.5357    8.7802 
226    3..5356    8.8185 
243    3.5355    8.8501 

5  41 

5  52 

6  04 

341    3.5348    8.9979 
352    3.5347    9.0118 
364                  9.0265 

35  36  36  37  38 
40  41  42  42  43 
45  46  47  48  49 

IX.     NATURAL  SIlfES  AND  COSINES. 


sin  n 

cos  n° 

vt" 

.0 

.1  .2  .3 

.4  .5  .6 

.7   .8   .9 

1.0 

(1 

0' 

0'   13'   18' 

24'  30'  30' 

42'   48'   54' 

CO' 

~0 

o.ouuu 

0017  003^0052 

0070  0087  OlOi 

0122  0140  0157 

"0l75 

"89 

78' 

1 

0175 

0192  020<;  0227 

0244  0262  0279 

0297  0314  0332 

0341; 

8S 

17 

2 

0349 

0366  0384  0401 

0419  0436  0454 

0471  0488  0506 

0523 

87 

17 

3 

0523 

0541  0558  0576 

0593  0610  0628 

0645  0663  0680 

069S 

86 

18 

4 

0(^.98'  0715  0732  0750 

0767  0785  0802 

0819  0837  0854 

0872 

85 

18 

5 

0872  0SS9  0906  0924 

0941  0958  0976 

0993  1011  1028 

1045 

84 

17 

6 

1045,  1063  lOSO  1097 

IIU  1132  1149 

1167  1184  1201 

1219 

83 

IS 

7 

1219  1236  1253  1271 

1288  1305  1323 

1.340  1357  1374 

1392 

82 

18 

8 

1392 

1409  1426  1444 

1461  1478  1495 

1513  1530  LS47 

1564 

81 

17 

9 

1564 

1 582  1.599  1616 

1633  1650  1668 

1685  1702  1719 

1736 

80 

17 

1010.1736 

1754  1771  1788 

1805  1822  1840 

1857  1874  1891 

1908 

79 

17 

11   190Si  1925  1942  1959 

1977  1994  2011 

2028  2045  2062 

2079 

78 

17 

12 

2079|  2096  2113  2130 

2147  2164  2181 

21982215  22.33 

2250 

77 

17 

13 

2250,  2267  22S4  2300J  2317  2334  2351 

2368  2385  2402 

2419 

76 

17 

14 

2419  2436  2453  2470 

2487  2504  2521 

2.538  25.54  2571 

2588 

75 

17 

15 

2588  2605  2622  2639 

2656  2672  2689 

2706  2723  2740 

2756 

74 

16 

10 

2756,  2773  2790  2807 

2823  2840  2857 

2874  2890  2907 

2924 

73 

17 

17 

29241  2940  2957  2974 

2990  3007  3024 

3040  3057  3074 

3090 

72 

16 

18 

3090  3107  3123  3140 

3156  3173  3190 

3206  3223  3239 

3256 

71 

17 

19 

3256  3272  3289  3305 

3322  3.338  ?,ZSS\  3371  3.387  3404 

3420 

70 

16 

20  ;0.3420;  3437  3453  3469|  3^86  3502  3518 

3535  35513567 

3584 

69 

17 

21 

35S4  3600  3616  3633 

3649  3665  3681 

3697  3714  3730 

3746  C8 

16 

oo 

3746,  3762  3778  3795 

.38113827  3843 

3859  3875  3891 

3907  67 

16 

23 

3907  3923  3939  3955 

3971  3987  4003 

4019  4035  4051 

4067  66 

16 

24 

4067,  4083  4099  4115  4131  4147  4163 

4179  4195  4210 

4226!  65 

16  . 

25 

4226,  4242  4258  4274  4289  4305  4321 

4337  4352  4368 

4384  64 

16 

26 

4384  4399  4415  4431  4446  4462  4478 

4493  4509  4524 

4540  63 

16 

27 

.  4540  4555  4571  4586  4602  4617  4633 

4648  4664  4679 

4695  62 

16 

28 

4695  4710  4726  4741  4756  4772  4787 

4802  4818  4833 

4848  61 

15 

2^ 

4848  4863  4879  4894  4909  4924  4939 

4955  4970  4985 

5000 

60 

15 

30  0.5000  5015  5030  5045 

5060  5075  5090 

5105  5120  5135 

5150 

59 

15 

31 

5150!  5 165  5 ISO  5 195 

5210  5225  5240 

5255  5270  5284 

5299 

58 

15 

32 

5299|  5314  5329  5344 

5358  5373  5388 

5402  5417  5432 

5446 

57 

14 

33 

5446  5461  5476  5490 

5.505  5519  5534 

5548  5563  5577 

5592 

56 

15 

34 

5592  5606  5621  5635 

5650  5664  5678 

5693  5707  5721 

5736 

55 

15 

35 

5736  5750  5764  57791  5793  5807  5821 

5835  5850  5864 

5878 

54 

14 

36 

5878  5892  5906  5920  5934  5948  5962 

5976  5990  6004 

6018 

53 

14 

37 

6018  6032  6046  6O60I  6074  6088  6101 

6115  6129  6143 

6157 

52 

14 

38 

6157  6170  6184  6198  6211  6225  6239 

6252  6266  6280 

6293 

lo 

13 

39 

6293,  6307  6320  6334;  6347  6361  6374 

6388  6401  6414 

6428 

ii 

40 

0.6428  6441  6455  6468  6481  6494  6508 

6521  6534  6547 

6561 

49 

14 

41 

6561  6574  6587  66O0!  6613  6626  6639 

6652  6665  6678 

6691 

48 

13 

42 

6691.  6704  6717  6730  6743  6756  6769 

6782  6794  6807 

6820 

47 

13 

43 

6820  6S33  6845  6858 

6871  6884  6896 

6909  6921  6934 

6947 

46 

13 

44 

6947  6959  6972  6984 

6997  7009  7022  7034  7046  7059i 

7071 

45 

12 

45 

7071  7083  7096  7108 

7120  7133  7145 

7157  7169  7181 

7193 

44 

12 

46 

7193  7206  7218  72301 

7242  7254  7266 

7278  7290  7302 

7314 

43 

12 

47 

7314!  7325  7337  7349 

7361  7373  7385 

7396  7408  7420 

7431 

42 

11 

48 

7431 

7443  7455  7466 

7478  7490  7501 

7513  7524  7.536 

7547 

41 

11 

49 

7547 

7559  7570  7581 

7593  7604  7615  7627  763S  7649 

7660 

40 

11 

ec 

54'  48'  42' 

3(5'   30'   24' 

18'  12'   6' 

0' 

1.0 

.9  .8  .7 

.6  .5  .4 

.3  .2  .1 

.0 

n" 

d 

IX.     NATUEAL   SIlfES   AND   COSINES. 


27 


sinn° 

COS  n° 

n° 

.0 

a  .2  .3 

.4   .5   .6 

.7   .8 

.9 

1.0 

d 

0' 

6'   13'  18' 

24'   30'   36' 

42'   48' 

54' 

60' 

50 

0.7660 

7672  7683  7694 

7705  7716  7727 

7738  7749 

776C 

7771 

"39 

IT 

51 

7771 

7782  7793  7804!  7815  7826  7837 

7848  7859 

7869 

7880  38 

11 

52 

7SS0 

7891  7902  7912'  7923  7934  7944 

7955  '7965 

7976 

7986^  37 

10 

53 

79S6 

7997  8007  8018  8028  8039  8049 

8059  8070 

8080 

8090'  36 

10 

54 

8090 

8100  81118121  8131  8141  8151 

8161  8171 

8181 

8192  35 

11 

55 

8192 

8202  8211  8221  8231  8241  8251 

8261  8271 

8281 

8290!  34 

9 

56 

8290 

8300  8310  8320  8329  8339  8348 

8358  8368 

8377 

8387 

33 

10 

57 

8387 

8396  8406  8415,8425  8434  8443 

8453  8^62 

8471 

8480 

32 

9 

58 

8480 

8490  8499  8508,8517  8526  8536 

8545  8554 

8563 

8572 

31 

9 

59 

8572 

8581  8590  8599  8607  8616  8625 

8634  8643 

8652 

8660 

30 

8 

60 

0.8669 

8669  8678  8686  8695  8704  8712 

8721  8729 

8738 

8746 

~2d 

T 

61 

8746 

8755  8763  8771 '8780  8788  8796 

8805  8813 

8821 

8829 

28 

8 

62 

8829 

8838  8846  8854!  8862  SS70  8878 

8886  8894 

8902 

8910|  27 

8 

63 

8910 

8918  8926  8934 

8942  8949  8957 

896i  8973 

8980 

8988  26 

S 

64 

8988 

8996  9003  9011 

9018  9026  9033 

9041  9048 

9056 

9063!  25 

7 

65 

9063 

9070  9078  9085 

9092  9100  9107 

9114  9121 

9128 

9135 

24 

7 

66 

9135 

9143  9150  9157 

9164  9171  9178 

9184  9191 

9198 

9205 

23 

7 

67 

9205 

9212  9219  9225 

9232  9239  9245 

9252  9259  9265 

9272 

22 

7 

68 

9272 

9278  9285  9291 

9298  9304  9311 

9317  9323 

9330 

9336 

21 

6 

69 

9336 

9342  9348  9354 

9361  9367  9373 

9379  9385  9391 

9397 

20 

6 

To 

0.9397 

9403  9409  9415 

9421  9426  9432 

9438  9444 

9449 

9455 

Id 

6 

71 

9455 

9461  9466  9472 

9478  9483  9489 

9494  9500 

9505 

9511 

18 

6 

72 

9511 

9516  95219527 

9532  9537  9542 

9548  9553 

9558 

9563 

17 

5 

73 

9563 

9568  9573  9578 

9583  9588  9593 

9598  9603 

9608 

9613 

16 

5 

74 

9613 

9617  9622  9627 

9632  9636  9641 

9646  9650 

9655 

9659 

15 

4 

75 

9659 

9664  9663  9673 

9677  9681  9686 

9690  9694 

9699 

9703 

14 

4 

76 

9703 

9707  97119715 

9720  9724  9728 

9732  9736  9740 

9744 

13 

4 

77 

9744 

9748  97519755 

9759  9763  9767 

9770  9774  9778 

9781 

12 

3 

78 

9781 

9785  9789  9792 

9796  9799  9803 

9806  9810  9813 

9816 

11 

3 

79 

9816 

9820  9823  9826 

9829  9833  9836 

9839  9842 

9845 

9848 

10 

3 

80 

0.9848  9851  9854  9857 

9860  9S63  9866 

9869  9871 

9874 

9877 

~~9 

T 

81 

9S77  9880  9882  9885 

9888  9890  9893 

9895  9898 

9900 

9903 

8 

3 

82 

9903 

9905  9907  9910 

9912  9914  9917 

9919  9921 

9923 

9925 

7 

2 

83 

9925 

9928  9930  9932 

9934  9936  9938 

9940  9942 

9943 

9945 

6 

2 

84 

9945 

9947  9949  9951 

9952  9954  9956 

9957  9959 

9960 

9962 

5 

2 

85 

9962 

9963  9965  9966 

9968  9969  9971 

9972  9973 

9974 

9976 

4 

2 

86 

9976  9977  9978  9979 

9980  9981  9982 

9983  9984 

9985 

9986 

3 

1 

87 

9986  9987  9988  9989 

9990  9990  9991 

9992  9993 

9993 

9994 

2 

1 

88 

9994:  9995  9995  9996 

9996  9997  9997!  9';97  9998  9998 

9998 

1 

0 

89 

9998  9999  9999  9999 

9999  1.000  1.000  J  1.000  1.000  1.000 

1.000 

0 

0 

60' 

54'  48'  42' 

36'   30'   24' 

18'   12' 

6' 

0' 

1.0 

.9  .8   .7 

.6   .5   .4 

.3   .2 

.1 

.0 

11° 

d 

12  3 

4  5  6 

7  8  9 

10 

11  12  13 

14  15  16 

17  18  19 

1 

0  0  1 

1  1  1 

1  1  2 

2 

2  2  2 

2  3  3 

3  3  3 

2 

0  1  1 

1  2  2 

2  3  3 

3 

4  4  4 

5  5  5 

6  6  6 

3 

1  1  2 

2  3  3 

4  4  5 

5 

6  6  7 

7    8  8 

9  9  10 

4 

1  1  2 

3  3  4 

5  5  6 

7 

7  8  9 

9  10  11 

11  12  13 

5 

1  2  3 

3  4  5 

6  7  8 

8 

9  10  11 

1 

I  U  U 

14  15  16 

28  X.   NATUEAL  TANGENTS  AND  COTANGENTS. 

tan  »»  cot  nP 


if 

.0 

0' 

.1  .2   .3 

0'   12'   18' 

.4  .5   .G 

24'  30'   36' 

.7  .8   .9 

42'  48'   54' 

1.0 

60' 

d 

0 
2 

:1 

4 
5 
0 
7 

8 
9 

0.0000 
0175 
0349 
0524 
0699 
0S75 
1051 
122S 
1405 
15S4 

0017  003i  0052 
0192  0209  0227 
0367  0384  a  J  02 
0.542  05.59  0577 
0717  07.34  0752 
0892  0910  092S 
1069  1086  1104 
1246  1263  1281 
1423  1441  14.59 
1602  1620  1638 

0070  0087  0105 
0244  0262  0279 
0419  0437  0454 
1  0594  0612  0629 
0769  0787  0805 
0945  0963  0981 
1122  1139  1157 
1299  1317  1334 
1477  1495  1512 
1655  167.3  1691 

0122  0140  0157 
0297  0314  0.332 
04  72  0489  0.507 
0647  0664  0682 
0822  0840  0857 
0998  1016  1033 
1175  1192  1210 
1352  1370  1388 
15.30  1548  1566 
1709  1727  1745 

0175 
0349 
0524 
0699 
0875 
1051 
1228 
1405 
1584 
1763 

:89 

'HS 
S7 
86 
85 
84 
83 
82 
81 
80 

18 
17 

!5 

18 
IS 
IS 
17 
18 
18 

10 
11 
12 
13 

u 

15 
1« 
17 
IS 
19 

0.1763 
1944 
2126 
2309 
2493 
2679 
2S67 
3057 
3249 
3443 

1781  1799  1817 
1962  1980  1998 
2144  2162  2180 
2327  2345  2364 
2512  25.30  2549 
2698  2717  2736 
2886  2905  2924 
3076  3096  3115 
3269  3288  3307 
3463  34S2  3502 

1835  1853  1871 
2016  2035  2053 
2199  2217  2235 
2382  2401  2419 
2568  2586  2605 
2754  2773  2792 
2943  2962  2981 
3134  3153  3172 
3327  3346  3365 
3522  3541  3561 

1890  1908  1926 
2071  2089  2107 
2254  2272  2290 
2438  2456  2475 
2623  2642  2661 
2811  2830  2849 
3000  3019  3038 
3191  3211  3230 
3385  3404  3424 
3581  3600  3620 

1944 
2126 
2309 
2493 
2679 
2867 
3057 
3249 
3443 
3640 

79 
78 
77 
76 
75 
74 
73 

71 

70 

18 
19 
19 
18 
18 
18 
19 
19 
19 
20 

20 
21 
2'* 
23 
24 
25 
2(i 
27 
2H 
29 

0.3640  3659  3679  3699 
3S39;  3S59  3879  3S99 
4040  4061  40S1  4101 
4245  4265  42S6  4307 
4452  4473  4494  4515 
4663'  46S4  4706  4727 
4S77  4S99  4921  4942 
5095  5117  5139  5161 
5317  5340  5362  53S4 
5543]  5566  5589  5612 

3719  3739  3759 
3919  3939  3959 
4122  4142  4163 
4327  4348  4369 
4536  4557  4578 
4748  4770  4791 
4964  4986  5008 
5184  5206  5228 
5407  5430  5452 
5635  5658  5681 

3779  3799  3819 
3979  4000  4020 
4183  4204  4224 
4390  4411  4431 
4599  4621  4642 
4813  4834  4856 
5029  5051  5073 
5250  5272  5295 
5475  5498  5520 
5704  5727  5750 

3839  60 
4040  6H 
4245  67 
4452  66 
4663165 
4877' 64 
5095  63 
5317  62 
5543  61 
5774  60 

20 
20 
21 
21 
21 
21 
22 
22 
23 
24 

30 
31 
32 
33 
34 
35 
3G 
37 
38 
39 

0.5774  5797  5820  5844 
6009  6032  6056  6080 
62491  6273  6297  6322 
6494!  6519  6544  6569 
6745;  67716796  6822 
7002  7028  7054  70S0 
7265  7292  7319  7346 
7536  7563  7590  7618 
7813  7841  7869  7S9S 
S09S  8127  8156  8185 

5867  5890  5914 
6104  6128  6152 
6346  6371  6395 
6594  6619  6644 
6847  6873  6899 
7107  7133  7159 
7373  7400  7427 
7646  7673  7701 
7926  7954  7983 
8214  8243  8273 

5938  5961  5985 
6176  6200  6224 
6420  6445  6469 
6669  6694  6720 
6924  6950  6976 
7186  7212  72.39 
7454  7481  7508 
7729  7757  7785 
8012  8040  8069 
8302  8332  8361 

6009 
6249 
6494 
6745 
7002 
7265 
7536 
7813 
8098 
8391 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

24 

25 
25 
25 
26 
26 
28 
28 
29 
30 

40 
41 
42 
43 
44 
45 
4G 
47 
48 
49 

0.8391 '  8421  8451  84S1 
8693  8724  8754  8785 
9004  90.36  9067  9099 
9325  9358  9391  9424 
9657|  9691  9725  9759 

1.0000  0035  0070  0105 
0355  0392  0428  0464 
0724' 0761  0799  083 7 
11061  1145  1184  1224 
1504  1544  1585  1626 

S51]  8541  8571 
8816  8847  8878 
9131  9163  9195 
9457  9490  9523 
9793  9827  9861 
01410176  0212 
0501  0538  05  7i 
0^5  0913  0951 
1263  1303  1343 
1667  1708  1750 

8601  8632  8662 
8910  8941  8972 
9228  9260  9293 
9556  9590  9623 
9896  9930  9965 
0247  0283  0319 
0612  0649  0686 
0990  1028  1067 
1383  1423  1463 
1792  1833  1875 

8693 
9004 
9325 
9657 
*0000 
0355 
0724 
1106 
1504 
1918 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

31 
32 
32 
34 
35 
36 
38 
39 
41 
43 

60'   54'  48'   42' 

1.0   .9  .8   .7 

36'  30'   24' 

.6  .5   .4 

18'   13'    6' 

.3   .2   .1 

0' 

.0 

n° 

d 

X. 


taii»*° 


NATURAL   TANGENTS   AND   COTANGENTS.        29 

cot»i-° 


n° 

.0 

.1      .2      .3   1 

.4      .5      .6 

.7      .8      .9 

1.0 

d 

50 

0' 

6'         13'       18'   1 

34'     30'       36' 

42'       48'      54' 

60' 

39 

^ 

1.1918 

1960  2002  2045  2088  2131  2174 

2218  2261  2305 

2349 

51 

2349 

2393  2437  2482:  2527  2572  2617 

2662  2708  2753 

2799 

38 

46 

52 

2799 

2846  2892  2938  2985  3032  3079 

3127  3175  3222 

3270 

37 

48 

53 

3270 

3319  3367  3416  3465  3514  3564 

3613  3663  3713 

3764 

36 

51 

54 

3764 

3814  3865  3916  3968  4019  4071 

4124  4176  4229 

4281 

35 

52 

55 

4281 

4335  4388  4442  4496  4550  4605 

4659  4715  4770 

4826 

34 

56 

56 

4826 

4882  4938  4994,  5051  5108  5166 

5224  5282  5340 

5399 

33 

59 

57 

5399 

5458  5517  5577  5637  5697  5757 

5818  5880  5941 

6003 

32;  62| 

58 

6003 

6066  6128  6191  6255  6319  6383 

6447  6512  6577 

6643 

31 

66 

59 
60 

6643 
1.7321 

6709  6775  6842  6909  6977  7045 

7113  7182  7251 

7321 

30 
29 

70 
74 

7391  7461  7532  7603  7675  7747 

7S20  7893  7966 

8040 

61 

8040 

8115  8190  8265  8341  8418  8495 

8572  8650  8728 

8807 

28 

79 

62 

8807 

8887  8967  9047  9128  9210  9292^  9375  9458  9542'  9626 

27 

84 

63 

9626 

9711  9797  9883  9970*0057*0145  *0233*0323*0413 

*0503 

26 

90 

64:2.0503 

0594  0686  0778  0872  0965  1060    1155  1251  1348 

1445 

25 1  97 

65 

1445 

1543  1642  1742  1842  1943  2045!  2148  2251  2355 

2460 

24105 

66 

2460 

2566  2673  2781  2889  2998*3109 

32^0  3332  3445 

3559 

23J114 

67 

3559 

3673  3789  3906  4023  4142  4262 

4383  4504  4627 

4751 

22!  124 

68 

4751 

4876  5002  5129'  5257  5386  5517 

5649  5782  5916 

6051 

21  135 

69 
70 

6051 

6187  6325  6464  6605  6746  6889 

7034  7179  7326i   7475 

20  149 
19164 

2.7475 

7625  7776  7929  8083  8239  8397 

8556  8716  8878    9042 

71 

9042 

9208  9375  9544  9714  9887*0061 

*0237*04 15*0595 1*07 7 7 

is:  182 

72 

3.0777 

0961  1146  1334  1524  1716  1910 

2106  2305  2506    2709 

17;203 

73 

2709 

2914  3122  3332:  3544  3759  3977 

4197  4420  4646    4874 

16J228 

74 

4874 

5105  5339  5576,  5816  6059  6305 

6554  6806  7062    7321 

15 

75 

7321 

7583  7848  8118!  8391  8667  8947 

9232  9520  9812*0108 

14 

76 

4.0108 

0408  0713  1022,  1335  1653  1976 

2303  2635  2972    3315 

13 

77 

3315 

3662  4015  4373' 4737  5107  5483 

5864  6252  6646!   7046 

12 

78 

7046 

7453  7867  82SS,  8716  9152  9594 

*0045*0504*0970:*1446 

11 

79 

80 

5.1446 

1929  2422  2924  3435  3955  4486 

.5026  55 

-8  6140|  6713 
f2*2432:*3138 

10 
9 

— 

5.6713 

7297  7894  8502  9124  9758*0405 

*1066*]7- 

81 

6.3138 

3859  4596  5350,  6122  6912  7720 

S548  9395*0264 i*l  1.54 

8 

827.1154 

2066  3002  3962  4947  5958  6996 

8062  91 58*0285!*  1443 

7 

83  8.1443 

2636  3863  5126  6427  7769  9152 

♦0579*2052*3572 

*5144 

6 

84 

9.514 

9.677  9.845  10.02  10.20  10.3910.58 

10.7810.9911.20 

11.43 

6 

85 

11.43 

11.6611.9112.16  12.43  12.7113.00 

13.30  13.62  13.95 

14.30 

4 

86 

14.30 

14.6715.0615.46  15.89  16.35  16.83 

17.34  17.8918.46  19.0S 

3 

87 

19.08 

19.74  20.45  21.20  22.02  22.90  23.86 

24.90  26.03  27.27  28.64 

2 

88 

28.64 

30.1431.82  33.69  35.80  38.19  40.92 

44.07  47.74  52.08 

57.29 

1 

89 

57.29 

63.66  71.62  81.85  95.49  114.6  143.2 

191.0  286.5  573.0 

0 

T 

1.0 

.9      .8      .7    1    .6      .5       .4 

.3      .2      .1 

.0 

20 

21  22  23 

24  25  26 

27  28  29 

30 

^1  32  33 

34  35  3637  38  39| 

T 

3 

4     4     4 

4     4     4 

5     5     5 

5 

5     5     6 

6    6    6 

6    6     7 

2 

7 

7     7    8 

8    8    9 

9    9  10 

10 

10  11  11 

11  12  12 

12  13  13 

3 

10 

11  11  12 

12  13  13 

14  14  15 

15 

16  16  17 

U''IS  18 

19  19  20 

4 

13 

14  15  15 

16  17  17 

IS  19  19 

20 

21  21  22 

23  23  24 

25  25  26 

5 

17 

18  IS  19 

20  21  22 

23  23  24 

25 

26  27  28 

28  29  30 

31  32  33 

40 

41  42  43 

44  45  46 

47  48  49 

50 

51  52  53 

54  55  56  57  58  59 

1 

7 

7     7     7 

7    8    8 

8     8    8 

8 

9    9    9 

9    9    9 [10  10  10 

2 

13 

14  14  14 

15  15  15 

16  16  16 

17 

17  17  18 

18  18  19 [  19  19  20 

3 

20 

21  21  22 

22  23  23 

24  24  25 

25 

26  26  27 

27  28  28  i  29  29  30 

4 

27 

27  28  29 

29  30  31 

31  32  33 

33 

34  35  35 

36  37  37   38  39  39 

5  133 

34  35  36 

37  38  38 

39  40  41 

42   43  43  44 

45  46  47 ' 48  48 

i^ 

30 


XI.     NATUEAL  SECANTS   AND   COSECANTS. 


sec  n° 

cosecn*^ 

u° 

.0 

.1  .2  .3 

.4   .5  .6 

.7   .8  .9 

1.0 

d 

O' 

C        12'  18' 

24'  :ui'    3<>' 

42'  48'  54' 

oooiTTooi  oooT 

CO' 

"0002 

89 

T 

0 

1.0000 

0000  0000  0000 

0000  0000  0001 

1 
2 

8 

0002  0002  0002  0003  1  0003  0003  0004 
0006  0007  0007  UU08  0009  0010  0010 
0014  OOli  U016  0U17  0018  0019  0020 

0004  0005  0006 
0011  0012  0013 
0021  0022  0023 

0006 
0014 

002^ 

88 
87 
80 

0 

1 
1 

4 
5 
(i 

0024  0026  0027  0028  0030  0031  0032  0034  0035  0037 
0038  i  0040  004 1  0W3  0045  0U46  0048  0050  0051  0053 
0055  1  0057  0059  0061  |  0063  0065  0067  ]  0069  0071  0073 

0038 
0055 
0075 

85 
84 
83 

1 
2 
2 

7 

8 
9 

0075 
0093 
0125 

0077  0079  0082  0084  00S6  00S9 
0101  0103  0106!  0108  0111  0114 
0127  01.^>0  0133i  0136  0139  0142 

0091  0093  0096 
0116  0119  0122 
0145  0148  0151 

0098 
0125 
0154 

82 
81 
80 

2 
3 
3 

10:i.0154 

0157  01610164 

0167  0170  0174 

0177  0180  0184 

0187 

79 

3 

11  01S7 

12  0223 
13'  02C3 

01910194  0198 
0227  02310235 
0267  0271  0276 

0201  0205  0209 
0239  0243  0247 
0280  0284  0288 

0212  0216  0220 
0251  0255  0259 
0293  0297  0302 

0223 
0263 
0306 

78 
77 
76 

3 
4 

4 

14  0306 
15!  ^-'^-^ 
l«i  0403 

03110315  0320 
0358  0363  0367 
0408  0413  (H 19 

0324  0329  0334 
0372  0377  0382 
0424  0429  0435 

0338  0343  03^8 
0388  0393  0398 
0440  04^6  0451 

0353 
0403 
0457 

75 
74 
73 

5 
5 

6 

17 
18 
19 

0457 
0515 
0576 

0^63  0468  0474 
05210527  0533 
0583  0589  0595 

0480  0485  0491 
0539  0545  0551 
0602  0608  0615 

0497  0503  0509 
0557  0564  0570 
0622  0628  0635 

0515 
0576 
0642 

72 
71 
70 

6 
6 

7 

20 

1.0642 

0649  0655  0662 

0669  0676  0683 

0690  0697  0704 

0711 

69 

7 

21 
22 
23 

0711 
0785 
0864 

0719  0726  0733 
0793  0801  0808 
0872  0880  0888 

0740  0748  0755 
0816  0824  0832 
0896  0904  0913 

0763  0770  0778 
0840  0848  0856 
0921  0929  0938 

0785 
0864 
0946 

68 
67 
66 

7 
8 
8 

241  0946 
25;  1034 
2«j  1126 

0955  0963  0972 
1043  1052  1061 
11361145  1155 

0981  0989  0998 
1070  1079  1089 
1164  1174  1184 

1007  1016  1025 
1098  1107  1117 
1194  1203  1213 

1034 
1126 
1223 

65 
64 
63 

9 
9 
10 

27  1223 

28  i  1326 

29  1434 

1233  1243  1253 
1336  1347  1357 
1445  1456  1467 

1264  1274  1284 
1368  1379  1390 
1478  1490  1501 

1294  1305  1315 
HOI  1412  1423 
1512  1524  1535 

1326 
1434 
1547 

62 
61 
60 

11 
11 
12 
12 

30  1.1547 

1559  1570  1582 

1594  1606  1618 

1630  1642  1654 

1666 

59 

31  1666  167916911703 

32  1792  11805.18181831 

33  '  1924  1937  1951  1964 

1716  1728  1741 
1844  1857  1870 
1978  1992  2006 

1753  1766  1779 
1883  1897  1910 
2020  2034  2048 

1792 
1924 
2062 

58 
57 
56 

13 
14 
14 

34 
35 
3G 

2062 
2208 
2361 

2076  2091  2105 
2223  2238  2253 
2376  2392  2408 

2120  2134  2149 
2268  2283  2299 
2424  2440  2456 

2163  2178  2193 
2314  2329  2345 
2472  24S9  2505 

2208 
2361 
2521 

55 
54 
53 

15 
16 
16 

37 
38 
39 

2521  2538  2554  2571 
2690  '  2708  2725  2742 
2868  '  2886  2904  2923 

2588  2605  2622 
2760  2778  2796 
2941  2960  2978 

2639  2656  2673 
2813  2831  2849 
2997  3016  .3035 

2690 
2868 
3054 

52 
51 
50 

17 
19 
19 

40 

1.3054,3073  3093  3112 

3131  31513171 

3190  3210  3230 

3250 

49 

20 

41 
42 
43 

3250  I  3270  3291  3311 
3456  3478  3499  3520 
3673  3696  3718  3741 

3331  3352  3373 
3542  3563  3585 
3763  3786  3809 

3393  .3414  3435 
3607  3629  3651 
3832  3855  3878 

3456 
3673 
3902 

48 
47 
46 

21 

22 
24 

44 
45 
46 

3902  1  3925  3949  3972 
4142,4167  4192  4217 
4396  4422  4448  4474 

3996  4020  4044 
4242  4267  4293 
4501  4527  4554 

4069  4093  4118 
4318  4344  4370 
4581  4608  4635 

4142 
4396 
4663 

45 
44 
43 

24 
26 

28 

47 

48 
49 

4663 
4945 
5243 

4690  4718  4746 
4974  5003  5032 
5273  5304  5335 

4774  4802  4830 
5062  5092  5121 
5366  5398  5429 

4859  4887  4916 
5151  5182  5212 
5461  5493  5525 

4945 
5243 
5557 

42 
41 
40 

29 
31 
32 

CO' 

1.0 

.'54'   48'  42' 

.9  .8  .7 

3<>'   ;jO'  24' 

.6   .5  .4 

18'   12'   G' 

.3  .2  .1 

0' 

.0 

n° 

d 

XI.     NATUEAL   SECANTS   AND   COSECANTS.  31 

sec  n°  cosec  n° 


1.5557 
5890 
6243 
6616 
7013 
7434 
78S3 
8361 
SS7I 
9416 


2.0000 
0627 
1301 
2027 
2812 
3662 
4586 
5593 
6695 
7904 


2.9238 
3.0716 
2361 
4203 
6280 
8637 
4.1336 
4454 
8097 
5.2408 
801 5.7588 
816.3925 
82  7.1853 
83'8.20S5 


9.567 
11.47 
14.34 
19.11 
28.65 
57.30 


.1   .2  .3 

6'   12'   18' 


5590 
5925 
6279 
6655 
7054 
7478 
7929 
8410 
8924 
9473 


5622  5655 
5959  5994 
6316  6353 
6694  6733 
7095  7137 
7522  7566 
7976  8023 
8460  8510 
8977  9031 
9530  9587 


0061 
0692 
1371 
2103 
2894 
3751 
4683 
5699 
681 1 
8032 


0122  0183 
0757  0824 
1441  1513 
2179  2256 
2976  3060 
3841  3931 
4780  4879 
5805  5913 
6927  7046 
8161  8291 


9379  9521 
0872  1030 
2535  2712 
4399  4598 
6502  6727 
8890  9147 
1627  1923 
4793  5137 
8496  8901 
2883  3367 


9665 
1190 
2891 
4799 
6955 
9408 
2223 
5486 
9313 
3860 


8164  8751  9351 
4637  5366  6111 
2757  3684  4635 
3238  4457  5711 
9.7289.89510.07 
11.7111.95  12.20 
14.7015.0915.50 
19.77  20.47  21.23 
30.1631.84  33.71 
63.66  71.62  81.85 


1.0   .9  .8   .7 


34'  30'  3G' 


5688  5721 
6029  6064 
6390  6427 
6772  6812 
7179  7221 
7610  7655 
8070  8118 
8561  8612 
9084  9139 
9645  9703 


5755 
6099 
6464 
6852 
7263 
7700 
8166 
8663 
9194 
9762 


0245 
0890 
1584 
2333 
3144 
4022 
4978 
6022 
7165 
8422 


0308  0371 
0957  1025 
1657  1730 
2412  2490 
3228  3314 
4114  4207 
5078  5180 
6131  6242 
7285  7407 
8555  8688 


9811  9957*0106 
1352  1515  1681 
3072  3255  3440 
5003  5209  5418 
7186  7420  7657 
9672  9939*0211 
2527  2837  3150 
5841  6202  6569 
9732*0159*0593 
4362  4874  5396 


.7   .8   .9   1.0 

43'   48'  54'   60' 


5788  5S22 
6135  6171 
6502  6540 
6892  6932 
7305  7348 
7745  7791 
8214  8263 
8714  8766 
9249  9304 
9821  9880 


5856 
6207 
6578 
6972 
7391 
7837 
8312 
8818 
9360 
9940 


0434 
1093 
1803 
2570 
3400 
4300 
5282 
6354 
7529 
8824 


0498  0562 
1162  1231 
1877  1952 
2650  2730 
3486  3574 
4395  4490 
5384  5488 
6466  6580 
7653  7778 
8960  9099 


*0256*0407*056] 
1848  2017  2188 
3628  3817  4009 
5629  5843  6060 
7897  8140  8387 

*04S6*0765*104S 
3469  3792  4121 
6942  7321  7706 

♦1034*1484*1942 
5928  6470  7023 


9963*0589  1227 
6874  7655  8454 
5611  6613  7642 
7004  8337  9711 
10.25  10.43  10.63 
12.47  12.75  13.03 
15.93  16.38  16.86 
22.04  22.93  23.88 
35.8138.20  40.93 
95.49114.6143.2 


.5      A 


1880  2546  3228 
9273*0112*0972 
8700  9787*0905 
n 129*2593*4105 
10.83  11.03  11.25 
13.34  13.65  13.99 
17.37  17.91  18.49 
24.92  26.05  27.29 
44.08  47.75  52.09 
191.0  286.5  573.0 


5890  39 
6243  38 
6616137 
7013j36 

7434  35 
7883  34 
8361 133 
8871 
9416 
=0000 


34 

36 

38 

41 

43 

46 

49 

321  53 

31 1  .56 

30  i  60 


0627 
1301 
2027 
2812 
3662 
4586 
5593 
6695 
7904 
^238  20 
^0716  19 
2361 
4203 
6280 
8637 
4336 
4454 
8097 
*2408 


_7588 
3925 
•■1853 
"2055 
^5668 
11.47 
14.34 
19.11 
28.65 
57.30 


.1       .0    n^   d 


60 


61  62  63  64  65  66  67  68  69l  70  71  72  73 


10  11 

11  11  11 

11  11  12 

)2 

12 

21  21 

21  22  22 

22  23  23 

23 

24 

31  32 

32  33  33 

34  34  35 

35 

36 

41  42 

43  43  44 

45  45  46 

47 

47 

52  53  53  54  55  : 56  57  58 

58 

59 

82  83J84  85  86,87  88  89 

90  91 

14  14 

14  14  14 

15  15  15 

15  15 

27  28 

28  28  29 

29  29  30 

30  130 

41  42 

42  43  43 

44  44  45 

45 

46 

55  55 

56  57  57 

58  59  59 

60 

61 

68  69 

70  71  72 

73  73  74 

75 

76 

12  12 
24  24 
36  37 
48  49 

60  61 
92  93 
15  16 
31  31 
46  47 

61  62 
77  78 


74  75  76 


12  13  13 
25  25  25 
37  38  38 
49  50  51 

62  63  63 
94  95  96 
16  16  16 
31  32  32 
47  48  48 

63  63  64 
78  79  80 


77  78  79 


13  13  13 
26  26  26 
39  39  40 
51  52  53 

64  65  66 
97  98  99 
16  16  17 
32  33  33 
49  49  50 

65  65  66 
81  82  S3 


32 


XII.    RADIANS. 


nir 

180 


n^ 

.0 

.1   .2   .3 

.4   .5   .0 

.7   .8   .9 

I> 

h  in 

10 

0.1745 

1763  1780  1798 

1815  1833  1850 

1868  1885  1902 

.0278 

0  40 

11 
13 

1920 
2094 
2269 

1937  1955  1972 
2112  2129  2147 
2286  2304  2321 

1990  2007  2025 
2164  2182  2199 
2339  2356  2374 

2042  2059  2077 
2217  2234  2251 
2391  2409  2426 

.0306 
.0333 
.0361 

0  44 

0  48 
0  52 

U 
15 
1« 

2443 
261S 
2793 

2461  2478  2496 
2635  2653  2670 
2810  2827  2845 

2513  2531  2548 
2688  2705  2723 
2862  2880  2897 

2566  2583  2601 
2740  2758  2775 
2915  2932  2950 

.0389 
.0417 
.0444 

0  56 

1  0 
1  4 

17 
IS 
19 

20 

2967 
3142 
3316 

0.3491 

2985  3002  3019 
3159  3176  3194 
3334  3351  3368 

3037  3054  3072 
3211  3229  3246 
3386  3403  3421 

3089  3107  3124 
3264  3281  3299 
3438  3456  3473 

.0472 
.0500 
.0528 

1  8 
1  12 
1  16 

3.i08  3526  3.543 

3560  3578  3595 

3613  3630  3648 

.0556 

1  20 

21 
2>> 

2:1 

3665 
3S40 
4014 

3683  3700  3718 
3857  3875  3892 
4032  4049  4067 

3735  3752  3770 
3910  3927  3944 
4084  4102  4119 

3787  3805  3822 
3962  3979  3997 
4136  4154  4171 

.0583 
.0611 
.0639 

124 

1  28 
1  32 

24 
25 
20 

41S9 
4363 
4.S3S 

4206  4224  4241 
4381  4398  4416 
4555  4573  4590 

4259  4276  4294 
4433  4451  4468 
4608  4625  4643 

4311  4328  4346.0667 
4485  4503  4520  .0694 
4660  4677  4695  .0722 

1  36 
1  40 
1  44 

27 

2S 
29 

30 

4712 
4SS7 
5061 

4730  4747  4765 
4904  4922  4939 
5079  5096  5114 

4782  4800  4817 
4957  4974  4992 
5131  5149  5166 

4835  4852  4869 
5009  5027  5044 
5184  5201  5219 

.0750 
.0778 
.0806 

1  48 
1  52 
1  56 

0.5236 

5253  5271  5288 

5306  5323  5341 

5358  5376  5393 

.0833 

2  0 

31 
32 
33 

5411 

55S5 
5760 

5428  5445  5463 
5603  5620  5637 
5777  5794  5812 

5480  5498  5515 
5655  5672  5690 
5829  5847  5864 

5533  55.50  5568 
5707  5725  5742 
5882  5899  5917 

.0861 
.0889 
.0917 

2  4 
2  8 
2  12 

34 
35 
30 

5934 
6109 
6283 

5952  5969  5986 
6126  6144  6161 
6301  6318  6336 

6004  6021  6039 
6178  6196  6213 
6353  6370  6388 

6056  6074  6091 
6231  6248  6266 
6405  6423  6440 

.0944 
.0972 
.1000 

2  16 
2  20 
2  24 

37 
38 
39 

40 

6458 
6632 
6807 

6475  6493  6510 
6650  6667  6685 
6824  6842  6859 

6528  6545  6562 
6702  6720  6737 
6877  6894  6912 

6580  6597  6615 
6754  6772  6789 
6929  6946  6964 

.1028 
.1056 
.1083 

2  28 
2  32 
2  36 

0.6981 

6999  7016  7034 

7051  7069  7086 

7103  7121  7138 

.1111 

2  40 

41 
42 
43 

7156 
7330 
7505 

7173  7191  7208 
7348  7365  7383 
7522  7540  7557 

7226  7243  7261 
7400  7418  7435 
7575  7592  7610 

7278  7295  7313 
7453  7470  7487 
7627  7645  7662 

.1139 
.1167 
.1194 

2  44 
2  48 

2  52 

44 
45 
40 

7679 
7854 
8029 

7697  7714  7732 
7871  7889  7906 
8046  8063  8081 

7749  7767  7784 
7924  7941  7959 
8098  8116  8133 

7802  7819  7837 
7976  7994  8011 
8151  8168  8186 

.1222 
.1250 
.1278 

2  56 

3  0 
3  4 

47 
4.S 
49 

5^ 

S203 
8378 
8552 

8221  8238  8255 
8395  8412  8430 
8570  8587  8604 

8273  8290  8308 
S447  8465  8482 
8622  8639  8657 

8325  8343  8360 
8500  8517  8535 
8674  8692  8709 

.1306 
.1333 
.1361 

3  8 
3  12 
3  16 

0.8727 

8744  8762  8779 

8796  8814  8831 

8849  8866  8884 

.1389 

3  20 

51 
52 
53 

8901 
9076 
9250 

8919  8936  8954 
9093  9111  9128 
9268  9285  9303 

8971  8988  9006 
9146  9163  9180 
9320  9338  9355 

9023  9041  9058 
9198  9215  9233 
9372  9390  9407 

.1417 
.1444 
.1472 

3  24 
3  28 
3  32 

54 
55 
50 

9425 
9599 
9774 

9442  9460  9477 
9617  9634  9652 
9791  9809  9826 

9495  9512  9529 
9669  9687  9704 
9844  9861  9879 

9547  9564  9582  .1500 
9721  9739  9756  .1528 
9896  9913  9931 '.1556 

3  36 
3  40 
3  44 

57 

5S 
59 

9948 

1.0123 

0297 

9966  9983*0001 
0140  0158  0175 
0315  0332  0350 

*0018*0036*0053|*O071*008S*O]05'.15S3 
0193  0210  0228  0245  0263  0280.1611 
0367  0385  0402  0420  0437  0455  .1639 

3  48 
3  52 
3  56 

180 


XII.    EADIANS. 


60 

.0 

.1   .2  .3 

.4   .5  .6 

.7   .8   .9   p 

h  111 

1.0472 

0489  0507  0524 

0542  0559  0577 

0594  0612  0629  .1667 

4  0 

61 
62 
63 

0647 
0821 
0996 

0664  0681  0699 
0S38  0856  0873 
1013  1030  1048 

0716  0734  0751 
0891  0908  0926 
1065  1083  1100 

0769  0786  0804 
0943  0961  0978 
1118  1135  1153 

.1694 
.1722 
.1750 

4  4 
4  8 
4  12 

64 
65 
66 

1170 
1345 
1519 

1188  1205  1222 
1362  1380  1397 
1537  1554  1572 

1240  1257  1275 
1414  1432  1449 
1589  1606  1624 

1292  1310  1327 
1467  1484  1502 
1641  1659  1676 

.1778 
.1806 
.1833 

4  16 

4  20 
4  24 

67 

68 
69 

70 

1694 
1868 
2043 

1711  1729  1746 
1886  1903  1921 
2060  2078  2095 

1764  1781  1798 
1938  1956  1973 
2113  2130  2147 

1816  1833  1851 
1990  2008  2025 
2165  2182  2200 

.1861 
.1889 
.1917 

4  28 
4  32 
4  36 

1.2217 

2235  2252  2270 

2287  2305  2322 

2339  2357  2374 

.1944 

4  40 

71 
72 
73 

2392 
2566 
2741 

2409  2427  2444 
2584  2601  2619 
2758  2776  2793 

2462  2479  2497 
2636  2654  2671 
2811  2828  2846 

2514  2531  2549 
2689  2706  2723 
2863  2881  2898 

.1972 
.2000 
.2028 

4  44 
4  48 
4  52 

74 
75 
76 

2915 
3090 
3265 

2933  2950  2968 
3107  3125  3142 
3282  3299  3317 

2985  3003  3020 
3160  3177  3195 
3334  3352  3369 

3038  3055  3073 
3212  3230  3247 
3387  3404  3422 

.2056 
.2083 
.2111 

4  56 

5  0 
5  4 

77 
78 
79 

3439 
3614 
3788 

3456  3474  3491 
3631  3648  3666 
3806  3823  3840 

3509  3526  3544 
3683  3701  3718 
3858  3875  3893 

3561  3579  3596 
3736  3753  3771 
3910  3928  3945 

.2139 
.2167 
.2194 

5  8 
5  12 
5  16 

80 

1.3963 

3980  3998  4015 

4032  4050  4067 

4085  4102  4120 

.2222 

5  20 

81 

82 
83 

4137 
4312 
4486 

4155  4172  4190 
4329  4347  4364 
4504  4521  4539 

4207  4224  4242 
4382  4399  4416 
4556  4573  4591 

4259  4277  4294 
4434  4451  4469 
4608  4626  4643 

.2250 
.2278 
.2306 

5  24 
5  28 
5  32 

84 
85 
86 

4661 
4835 
5010 

4678  4696  4713 
4853  4870  4888 
5027  5045  5062 

4731  4748  4765 
4905  4923  4940 
5080  5097  5115 

4783  4800  4818 
4957  4975  4992 
5132  5149  5167 

.2333 
.2361 
.2389 

5  36 

5  40 
5  44 

87 
88 
89 

5184 
5359 
5533 

5202  5219  5237 
5376  5394  5411 
5551  5568  5586 

5254  5272  5289 
5429  5446  5464 
5603  5621  5638 

5307  5324  5341 
5481  5499  5516 
5656  5673  5691 

.2417 
.2444 
.2472 

5  48 
5  52 
5  56 

90 

1.5708 

5725  5743  5760 

5778  5795  5813 

5830  5848  5865 

.2500 

6  0 

91 
92 
98 

5S82 
6057 
6232 

5900  5917  5935 
6074  6092  6109 
6249  6266  6284 

5952  5970  5987 
6127  6144  6162 
6301  6319  6336 

6005  6022  6040 
6179  6197  6214 
6354  6371  6389 

.2528 
.2556 
.2583 

6  4 
6  8 
6  12 

94 
95 
96 

6406 
6581 
6755 

6424  6441  6458 
6598  6616  6633 
6773  6790  6808 

6476  6493  6511 
6650  6668  6685 
6825  6842  6860 

6528  6546  6563 
6703  6720  6738 
6877  6895  6912 

.2611 
.2639 
.2667 

6  16 
6  20 
6  24 

97 

98 
99 

6930 
7104 
7279 

6947  6965  6982 
7122  7139  7157 
7296  7314  7331 

7000  7017  7034 
7174  7191  7209 
7349  7366  7383 

7052  7069  7087 
7226  7244  7261 
7401  7418  7436 

.2694 

.2722 
.2750 

6  28 
6  32 
6  36 

T 

2 
3 

0    12   3 

4   5   6 

7    8   9 

P 

s 

4 
S 
12 

.00  2909!  3200  3491  3782 
.00  5818  6109  6400  6690 
.00  8727  9018  9308  9599 

4072  4363  4654 
6981  7272  7563 
9890*0181*0472 

4945  5236  5527 
7854  8145  8436 
*0763*1054*1345 

.000  0463 
0926 
1389 

4 
5 
6 

.01  163 
.01  454 
.01  745 

6  1926  2217  2508 
4  4835  5126  5417 
3  7744  8035  8326 

2799  3090  3381 
5708  5999  6290 
8617  8908  9199 

3672  3963  4254 
6581  6872  7162 
9490  9780*0071 

1852 
2315 

2778 

16 
20 
24 

7 
8 
9 

.02  036 
.02  327 
.02  618 

2  0653  0944  1235 
1  3562  3853  4144 
d   6471  6762  7053 

1526  1817  2108 
4435  4725  5016 
7343  7634  7925 

2398  2689  2980 
5307  5598  5SS9 
8216  8507  8798 

3241 
3704 
4167 

28 
32 
36 

34 


XIII.     EEOIPROOALS. 


n 

0 

1   2   3 

4   5   6 

7   8   9 

d 

1.0 

1.0000 

9901  9804  9709 

9615  9524  9434 

9346  9259  9174 

83 

1.1 
1.- 
l!ii 

0.9091 
S333 
7692 

9009  8929  8850 
8264  8197  8130 
7634  7576  7519 

8772  8696  8621 
8065  8000  7937 
7463  7407  7353 

8547  8475  8403 
7874  7813  7752 
7299  7246  7194 

70 
60 
51 

1.4 
1.5 
l.G 

7143 
6667 
6250 

7092  7042  6993 
6623  6579  6536 
6211  6173  6135 

6944  6897  6849 
6494  6452  6410 
6098  6061  6024 

6803  6757  6711 
6369  6329  6289 
5988  5952  5917 

44 
39 
35 

1.7 
1.8 
1.9 

5SS2 
5556 
5263 

5848  5814  5780 
5525  5495  5464 
5236  5208  5181 

5747  5714  5682 
5435  5405  5376 
5155  5128  5102 

5650  5618  5587 
5348  5319  5291 
5076  5051  5025 

31 
28 
25 

2.0 

0.5000 

4975  4950  4926 

4902  4878  4854 

4831  4808  4785 

23 

2.1 
2.2 
2.3 

4762 
4545 
434S 

4739  4717  4695 
4525  4505  4484 
4329  4310  4292 

4673  4651  4630 
4464  4444  4425 
4274  4255  4237 

4608  4587  4566 
4405  4386  4367 
4219  4202  4184 

21 
19 
17 

2.4 
2.5 

2.0 

4167 
4000 
3S46 

4149  4132  4115 
3984  3968  3953 
3831  3817  3S02 

4098  4082  4065 
3937  3922  3906 
3788  3774  3759 

4049  4032  4016 
3891  3876  3861 
3745  3731  3717 

16 

15 
13 

2.7 

2.8 
2.9 

3704 
3571 
3448 

3690  3676  3663 
3559  3546  3534 
3436  3425  3413 

3650  3636  3623 
3521  3509  3497 
3401  3390  3378 

3610  3597  3584 
3484  3472  3460 
3367  3356  3344 

13 
12 
11 

3.0 

03333 

3322  3311  3300 

3289  3279  3268 

3257  3247  3236 

10 

3.1 
3.2 
3.3 

3226 
3125 
3030 

3215  3205  3195 
3115  3106  3096 
3021  3012  3003 

3185  3175  3165 
3086  3077  3067 
2994  2985  2976 

3155  3145  3135 
3058  3049  3040 
2967  2959  2950 

10 
10 
9 

3.4 
3.5 
3.6 

2941 
2857 
2778 

2933  2924  2915 
2849  2841  2833 
2770  2762  2755 

2907  2899  2890 
2825  2817  2809 
2747  2740  2732 

2882  2874  2865 
2801  2793  2786 
2725  2717  2710 

8 

8 

7 

3.7 
3.8 
3.9 

2703 
2632 
2564 

2695  2688  2681 
2625  2618  2611 
2558  2551  2545 

2674  2667  2660 
2604  2597  2591 
2538  2532  2525 

2653  2646  2639 
2584  2577  2571 
2519  2513  2506 

7 
7 
6 

4.0 

0.2500 

2494  2488  2481 

2475  2469  2463 

2457  2451  2445 

6 

4.1 
4.2 
4.3 

2439 
2381 
2326 

2433  2427  2421 
2375  2370  2364 
2320  2315  2309 

2415  2410  2404 
2358  2353  2347 
2304  2299  2294 

2398  2392  2387 
2342  2336  2331 
2288  2283  2278 

6 

5 
5 

4.4 
4.5 
4.6 

2273 
2222 
2174 

226S  2262  2257 
2217  2212  2208 
2169  2165  2160 

2252  2247  2242 
2203  2198  2193 
2155  2151  2146 

2237  2232  2227 
2188  2183  2179 
2141  2137  2132 

5 
5 
4 

4.7 

4.8 
4.9 

2128 
2083 
2041 

2123  2119  2114 
2079  2075  2070 
2037  2033  2028 

2110  2105  2101 
2066  2062  2058 
2024  2020  2016 

2096  2092  20SS 
2053  2049  2045 
2012  2008  2004 

5 
4 
4 

5.0 

0.2000 

1996  1992  1988 

1984  1980  1976 

1972  1969  1965 

4 

5.1 
5.2 
5.3 

1961 
1923 

18S7 

1957  1953  1949 
1919  1916  1912 
1883  1880  1876 

1946  1942  1938 
1908  1905  1901 
1873  1869  1866 

1934  1931  1927 
1898  1894  1890 
1862  1859  1855 

4 
3 
3 

5.4 
5.5 
5.6 

1852 
1818 
1786 

1848  1845  1842 
1815  1812  1808 
1783  1779  1776 

1838  1835  1832 
1805  1802  1799 
1773  1770  1767 

1828  1825  1821 
1795  1792  1789 
1764  1761  1757 

3 
3 
3 

5.7 

5.8 
5.9 

1754 
1724 
1695 

1751  1748  1745 
1721  1718  1715 
1692  1689  1686 

1742  1739  1736 
1712  1709  1706 
1684  1681  1678 

1733  1730  1727 
1704  1701  1698 
1675  1672  1669 

3 
3 
2 

1 

n 

XIII.  EEOIPROOALS. 

35 

n 

0 

12   3 

4   5   6 

7 

8   9 

d 

6.0 

0.1667 

1664  1661  1658 

1656  1653  1650 

1647  1645  1642 

3 

6.1 
6.2 
6.3 

1639 
1613 

1587 

1637  1634  1631 
1610  1608  1605 
1585  1582  1580 

1629  1626  1623 
1603  1600  1597 
1577  1575  1572 

1621 
159i 
157C 

1618  1616 
1592  1590 
1567  1565 

3 
3 

2 

6.4 
6.5 
6.6 

1563 
1538 
1515 

1560  1558  1555 
1536  1534  1531 
1513  1511  1508 

1553  1550  1548 
1529  1527  1524 
1506  1504  1502 

154C 

1522 
1495 

1543  1541 
1520  1517 
1497  1495 

3 

2 
2 

6.7 
6.8 
6.9 

1493 
1471 
1449 

1490  1488  1486 
1468  1466  1464 
1447  1445  1443 

1484  1481  1479 
1462  1460  1458 
1441  1439  1437 

1477 
1456 
1435 

1475  1473 
1453  1451 
1433  1431 

2 
2 
2 

7.0 

0.1429 

1427  1425  1422 

1420  1418  1416 

1414 

1412  1410 

2 

7.1 
7.2 
7.3 

1408 
13Sv 
1370 

1406  1404  1403 
1387  1385  1383 
1368  1366  1364 

1401  1399  1397 
1381  1379  1377 
1362  1361  1359 

1395 
1376 
1357 

1393  1391 
1374  1372 
1355  1353 

2 
2 
2 

7.4 
7.5 

7.6 

1351 
1333 
1316 

1350  134S  1346 
1332  1330  1328 
1314  1312  1311 

1344  1342  1340 
1326  1325  1323 
1309  1307  1305 

1339 
1321 
1304 

1337  1335 
1319  1318 
1302  1300 

2 
2 

7.7 
7.8 
7.9 

1299 
1282 
1266 

1297  1295  1294 
1280  1279  1277 
1264  1263  1261 

1292  1290  1289 
1276  1274  1272 
1259  1258  1256 

1287  1285  1284 
1271  1269  1267 
1255  1253  1252 

2 

1 
2 

8.0 

0.1250 

1248  1247  1245 

1244  1242  1241 

1239  1238  1236 

1 

8.1 

8.2 
8.3 

1235 
1220 
1205 

1233  1232  1230 
1218  1217  1215 
1203  1202  1200 

1229  1227  1225 
1214  1212  1211 
1199  1198  1196 

1224 
1209 
1195 

1222  1221 
1208  1206 
1193  1192 

1 

1 
2 

8.4 
8.5 
8.6 

1190 
1176 
1163 

1189  1188  11S6 
1175  1174  1172 
1161  1160  1159 

1185  1183  1182 
1171  1170  1168 
1157  1156  1155 

1181 
1167 
1153 

1179  1178 
1166  1164 
1152  1151 

2 
1 
2 

8.7 
8.8 
8.9 

1149 
1136 
1124 

1148  1147  1145 
1135  1134  1133 
1122  1121  1120 

1144  1143  1142 
1131  1130  1129 
1119  1117  1116 

1140 
1127 
1115 

1139  1138 
1126  1125 
1114  1112 

2 

1 
1 

9.0 

0.1111 

1110  1109  1107 

1106  1105  1104  1103 

1101  1100 

1 

9.1 
9.2 
9.3 

1099 
1087 
1075 

1098  1096  1095 
1086  1085  1083 
1074  1073  1072 

1094  1093  1092  1091 
1082  1081  1080  i  1079 
1071  1070  1068  !  1067 

1089  1088 
1078  1076 
1066  1065 

1 
1 
1 

9.4 
9.5 
9.6 

1064 
1053 
1042 

1063  1062  1060 
1052  1050  1049 
1041  1040  1038 

1059  1058  1057  I  1056 
1048  1047  1046  ;  1045 
1037  1036  1035  ,  1034 

1055  1054 
1044  1043 
1033  1032 

1 
1 
1 

9.7 
9.8 
9.9 

1031 
1020 
1010 

1030  1029  1028 
1019  1018  1017 
1009  1008  1007 

1027  1026  1025  1024 
1016  1015  1014  !  1013 
1006  1005  1004  !  1003 

1022  1021 
1012  1011 
1002  1001 

1 
1 
1 

2 

3 

4  5  6 

7   8   9 

11   12   13 

14   15  16 

1 
2 
3 

.5  . 

(3) 
(6) 

.25  .2.1(6) 
.5  .4.(3) 
.75  .6  .5 

.1429  .125 
.2857  .25 
.4286  .375 

.(2) 
.(3) 

.(09) 
.(IS) 
.(27) 

08(3)  .0769 
1(6)  .1538 
25   .2308 

.0714  .0(6)  .0625 
.1429.1(3)  .125 
.2143 .2   .1875 

4 
5 
6 

.8  .(6) 
.8(3) 

.5714  .5 
.7143  .625 
.8571  .75 

•(4) 
.(5) 
.(6) 

.(36) 
•(45) 
.(54) 

(3)   .3077 
41(6)  .3846 
5    .4615 

.2857  .2(6)  .25 
.3571.(3)  .3125 
.4286 .4   .375 

7 

8 
9 

.875 

:S 

.(63) 

.(72) 
■(SI) 

58(3)  .5385 
(6)   .6154 
75   .6923 

.5   .4(6)  .4375 
.5714.5(3)  .5 
.6429 .6   .5625 

3G 


XIV.     SQUARES. 


u^ 


n 

0 

1         2       3 

4        6       6 

7          8        9 

d 

1.0 

1.000 

1.020  1.040  1.061 

1.082  1.103  1.124 

1.145  1.166  1.188 

22 

1.1 
1.2 
1.3 

1.210 
1.440 
1.690 

1.232  1.254  1.277 
1.464  1.488  1.513 
1.716  1.742  1.769 

1.300  1.323  1.346 
1.538  1..S63  1.588 
1.796  1.823  1.850 

1.369  1.392  1.416 
1.613  1.638  1.664 
1.877  1.904  1.932 

24 

26 

28 

1.4 
1.5 
1.6 

l.%0 
2.250 
2..S60 

1.9SS  2.016  2.045 
2.280  2.310  2.341 
2.592  2.624  2.657 

2.074  2.103  2.132 
2.372  2.403  2.434 
2.690  2.723  2.756 

2.161  2.190  2.220 
2.465  2.496  2.528 
2.789  2.822  2.856 

30 
32 
34 

1.7 

1.8 
1.9 

2.S90 
3.240 
3.610 

2.924  2.958  2.993 
3.276  3.312  3.349 
3.648  3.686  3.725 

3.028  3.063  3.098 
3.386  3.423  3.460 
3.764  3.803  3.842 

3.133  3.16S  3.204 
3.497  3.534  3.572 
3.881  3.920  3.960 

36 
38 
40 

2.0 

4.000 

4.040  4.080  4.121 

4.162  4.203  4.244 

4.285  4.326  4.368 

42 

2.1 
2.2 
2!3 

4.410 
4.S40 
5.290 

4.452  4.494  4.537 
4.884  4.928  4.973 
5.336  5.382  5.429 

4.580  4.623  4.666 
5.018  5.063  5.108 
5.476  5.523  5.570 

4.709  4.752  4.796 
5.153  5.198  5.244 
5.617  5.664  5.712 

44 
46 
48 

2.4 
2.5 
2.G 

5.760 
6.250 
6.760 

5.808  5.856  5.905 
6.300  6.350  6.401 
6.812  6.864  6.917 

5.954  6.003  6.052 
6.452  6.503  6.554 
6.970  7.023  7.076 

6.101  6.150  6.200 
6.605  6.656  6.708 
7.129  7.182  7.236 

50 
52 
54 

2.7 

2.8 
2.9 

7.290 
7.S40 
8.410 

7.344  7.398  7.453 
7.896  7.952  8.009 
8.468  8.526  8.585 

7.508  7.563  7.618 
8.066  8.123  S.ISO 
8.644  8.703  8.762 

7.673  7.728  7.784 
8.237  8.294  8.352 
8.821  8.880  8.940 

56 
58 
60 

3.0 

9.000 

9.060  9.120  9.181 

9.242  9.303  9.364 

9.425  9.486  9.548 

62 

3.1 
3.2 
3.3 

9.610 
10.24 
10.89 

9.672  9.734  9.797 
10.30  10.37  10.43 
10.96  11.02  11.09 

9.S60  9.923  9.986 
10.50  10.56  10.63 
11.16  11.22  11.29 

10.05  10.11  10.18 
10.69  10.76  10.82 
11.36  11.42   11.49 

6 

7 
7 

3.4 
3.5 
3.6 

11.56 
12.25 
12.96 

11.63  11.70  11.76 
12.32  12.39  12.46 
13.03  13.10  13.18 

11.83  11.90  11.97 
12.53  12.60  12.67 
13.25  13.32  13.40 

12.04  12.11  A2.18 
12.74  12.82  12.89 
13.47  13.54  13.62 

7 

7 
7 

3.7 
3.8 
3.9 

13.69 
14.44 
15.21 

13.76  13.84  13.91 
14.52  14.59  14.67 
15.29  15.37  15.44 

13.99  14.06  14.14 
14.75  14.82  14.90 
15.52  15.60  15.68 

14.21  14.29  14.36 
14.98  15.05  15.13 
15.76  15.84  15.92 

8 
8 
8 

4.0 

16.00 

16.08  16.16  16.24 

16.32  16.40  16.48 

16.56  16.65  1673 

8 

4.1 
4.2 
4.3 

16.81 
17.64 
18.49 

16.89  16.97  17.06 
17.72  17.81  17.89 
18.58  18.66  18.75 

17.14  17.22  17.31 
17.98  18.06  18.15 
18.84  18.92  19.01 

17.39  17.47  17.56 
18.23  18.32  18.40 
19.10  19.18  19.27 

8 
9 
9 

4.4 
4.5 
4.6 

19.36 
20.25 
21.16 

19.45  19.54  19.62 
20.34  20.43  20.52 
21.25  21.34  21.44 

19.71  19.80  19.89 
20.61  20.70  20.79 
21.53  21.62  21.72 

19.98  20.07  20.16 
20.88  20.98  21.07 
21.81  21.90  22.00 

9 
9 

9 

4.7 

4.8 
4.9 

22.09  1  22.18  22.28  22.37 
23.04    23.14  23.23  23.33 
24.01  j  24.11  24.21  24.30 

22.47  22.56  22.66 
23.43  23.52  23.62 
24.40  24.50  24.60 

22.75  22.85  22.94 
23.72  23.81  23.91 
24.70  24.80  24.90 

10 
10 
10 

5.0 

25.00  ,  25.10  25.20  25.30 

25.40  25.50  25.60 

25.70  25.81  25.91 

10 

6.1 
5.2 
6.3 

26.01     26.11  26.21  26.32 
27.04  i  27.14  27.25  27.35 
28.09  1  28.20  28.30  28.41 

26.42  26.52  26.63 
27.46  27.56  27.67 
28.52  28.62  28.73 

26.73  26.83  26.94 
27.77  27.88  27.98 
28.84  28.94  29.05 

10 
11 
11 

6.4 
6.5 
5.6 

29.16    29.27  29.38  29.48 
30.25  !  30.36  30.47  30.58 
31.36  i  31.47  31.58  31.70 

29.59  29.70  29.81 
30.69  30.80  30.91 
31.81  31.92  32.04 

29.92  30.03  30.14 
31.02  31.14  31.25 
32.15  32.26  32.38 

11 
11 
11 

5.7 

5.8 
5.9 

32.49  1  32.60  32.72  32.83 
33.64  1  33.76  33.87  33.99 
.34.81  1  34.93  35.05  35.16 

32.95  33.06  33.18 
34.11  34.22  34.34 
35.28  35.40  35.52 

33.29  33.41  33.52 
34.46  34.57  34.69 
35.64  35.76  35.88 

12 
12 
12 

n^ 


XIV.     SQUAEES. 


37 


n 

0 

12        3 

4         5        6 

7         8        9 

d 

6.0 

36.00 

36.12  36.24  36.36 

36.48  36.60  36.72 

36.84  36.97  37.09 

12 

6.1 
6.2 
6.3 

37.21 
38.44 
39.69 

37.33  37.45  37.58 
38.56  38.69  38.81 
39.82  39.94  40.07 

37.70  37.82  37.95 
38.94  39.06  39.19 
40.20  40.32  40.45 

38.07  38.19  38.32 
39.31  39.44  39.56 
40.58  40.70  40.83 

12 
13 
13 

6.4 
6.5 
6.6 

40.96 
42.25 
43.56 

41.09  41.22  41.34 
42.38  42.51  42.64 
43.69  43.82  43.96 

41.47  41.60  41.73 
42.77  42.90  43.03 
44.09  44.22  44.36 

41.86  41.99  42.12 
43.16  43.30  43.43 
44.49  44.62  44.76 

13 
13 
13 

6.7 
6.8 
6.9 

44.89 
46.24 
47.61 

45.02  45.16  45.29 
46.38  46.51  46.65 
47.75  47.89  48.02 

45.43  45.56  45.70 
46.79  46.92  47.06 
48.16  48.30  48.44 

45.83  45.97  46.10 
47.20  47.33  47.47 
48.58  48.72  48.86 

14 
14 
14 

7.0 

49.00 

49.14  49.28  49.42 

49.56  49.70  49.84 

49.98  50.13  50.27 

14 

7.1 
7.2 
7.3 

50.41 
51.84 
53.29 

50.55  50.69  50.84 
51.98  52.13  52.27 
53.44  53.58  ,53.73 

50.98  51.12  51.27 
52.42  52..S6  52.71 
53.88  54.02  54.17 

51.41  51.55  51.70 
52.85  53.00  53.14 
54.32  54.46  54.61 

14 
15 
15 

7.4 
7.5 
7.6 

54.76 
56.25 
57.76 

54.91  55.06  55.20 
56.40  56.55  56.70 
57.91  58.06  58.22 

55.35  55.50  55.65 
56.85  57.00  57.15 
58.37  58.52  58.68 

55.80  55.95  56.10 
57.30  57.46  57.61 
58.83  58.98  59.14 

15 
15 
15 

7.7 

7.8 
7.9 

59.29 
60.84 
62.41 

59.44  59.60  59.75 
61.00  61.15  61.31 
62.57  62.73  62.88 

59.91  60.06  60.22 
61.47  61.62  61.78 
63.04  63.20  63.36 

60.37  60.53  60.68 
61.94  62.09  62.25 
63.52  63.68  63.84 

16 
16 
16 

8.0 

64.00 

64.16  64.32  64.48 

64.64  64.80  64.96 

65.12  65.29  65.45 

16 

8.1 
8.2 
8.3 

65.61 
67.24 
68.89 

65.77  65.93  66.10 
67.40  67.57  67.73 
69.06  69.22  69.39 

66.26  66.42  66.59 
67.90  68.06  68.23 
69.56  69.72  69.89 

66.75  66.91  67.08 
68.39  68.56  68.72 
70.06  70.22  70.39 

16 
17 
17 

8.4 
8.5 
8.6 

70.56 
72.25 
73.96 

70.73  70.90  71.06 
72.42  72.59  72.76 
74.13  74.30  74.48 

71.23  71.40  71.57 
72.93  73.10  73.27 
74.65  74.82  75.00 

71.74  71.91  72.08 
73.44  73.62  73.79 
75.17  75.34  75.52 

17 
17 

17 

8.7 
8.8 
8.9 

75.69 
77.44 
79.21 

75.86  76.04  76.21 
77.62  77.79  77.97 
79.39  79.57  79.74 

76.39  76.56  76.74 
78.15  78.32  78.50 
79.92  80.10  80.28 

76.91  77.09  77.26 
78.68  78.85  79.03 
80.46  80.64  80.82 

18 
18 
18 

9.0 

81.00 

81.18  81.36  81.54 

81.72  81.90  82.08 

82.26  82.45  82.63 

18 

9.1 
9.2 
9.3 

82.81 
84.64 
86.49 

82.99  83.17  83.36 
84.82  85.01  85.19 
86.68  86.86  87.05 

83.54  83.72  83.91 
85.38  85.56  85.75 
87.24  87.42  87.61 

84.09  84.27  84.46 
85.93  86.12  86.30 
87.80  87.98  88.17 

18 
19 
19 

9.4 
9.5 
9.6 

88.36 
90.25 
92.16 

88.55  88.74  88.92 
90.44  90.63  90.82 
92.35  92.54  92.74 

89.11  89.30  89.49 
91.01  91.20  91.39 
92.93  93.12  93.32 

89.68  89.87  90.06 
91.58  91.78  91.97 
93.51  93.70  93.90 

19 
19 
19 

9.7 
9.8 
9.9 

94.09 
96.04 
98.01 

94.28  94.48  94.67 
96.24  96.43  96.63 
98.21  98.41  98.60 

94.87  95.06  95.26 
96.83  97.02  97.22 
98.80  99.00  99.20 

95.45  95.65  95.84 
97.42  97.61  97.81 
99.40  99.60  99.80 

20 
20 
20 

n 

.0 

.1      .2      .3 

.4      .5      .6 

.7      .8       .9 

d 

1 
2 
3 

1.000 
0.250 
0.111 

*826  *694  *592 
227     207     189 
104  *977  *91S 

*510  *444  *391 

174     160     148 

*S65  *816  *772 

*346  *309  *277 

137     128     119 

*730  *693  *657 

27 

8 

32 

4 
5 
6 

0.0  625 
0.0  400 
0.0  278 

595     567     541 
384     370     356 
269     260    252 

517    494    473 
343     331     319 
244     237     230 

453     434    416 
308    297     287 
223     216    210 

16 
9 
6 

7 
8 
9 

0.0  204 
0.0  156 
0.0  123 

198     193     188 
152     149     145 
121     118     116 

183     178     173 
142     138     135 
113     111     109 

169     164     160 
132     129     126 
106     104     102 

4 
3 
2 

38 


XV.     CUBES. 


11-^ 


n 

0 

1         2       8 

4         5        6 

7         8        9 

d 

1.0 

1.000 

1.030  1.061  1.093 

1.125  1.158  1.191 

1.225  1.260  1.295 

36 

1.1 
1  '' 
l!3 

1.331 

1.728 
2.197 

1.368  1.405  1.443 
1.772  1.816  1.861 
2.248  2.300  2.353 

1.482  1.521  1.561 
1.907  1.953  2.000 
2.406  2.460  2.515 

1.602  1.643  1.685 
2.048  2.097  2.147 
2.571  2.628  2.686 

43 
50 
58 

1.4 
1.5 
1.6 

2.744  2.803  2.863  2.924 
3.375  3.443  3.512  3.582 
4.096  4.173  4.252  4.331 

2.986  3.049  3.112 
3.652  3.724  3.796 
4.411  4.492  4.574 

3.177  3.242  3.308 
3.870  3.944  4.020 
4.657  4.742  4.827 

67 
76 
86 

1.7 

l.H 
1.9 

2^ 

4.913   5.000  5.0S8  5.178 
5.832:  5.930  6.029  6.128 
6.859^  6.968  7.078  7.189 

5.268  5.359  5.452 
6.230  6.332  6.435 
7.301  7.415  7.530 

5.545  5,640  5.735 
6.539  6.645  6.751 
7.645  7.762  7.881 

97 
108 
119 

8.000  8.121  8.242  8.365 

8.490  8.615  S.742 

8.870  8.999  9.129 

132 

2.1 
2.2 
2!3 

9.261:9.394  9.528  9.664 
10.648   10.79  10.94  11.09 
12.167   12.33  12.49  12.65 

9.800  9-.938  10.08 
11.24  11.39  11.54 
12.81  12.98  13.14 

10.22  10.36  10.50 
11.70  11.85  12.01 
13.31  13.48  13.65 

15 
16 
17 

2.4'  13.824]  14.00  14.17  14.35 

2.5  15.625   15.81  16.00  16.19 

2.6  17.576  17.78  17.98  18.19 

14.53  14.71  14.89 
16..39  16,58  16.78 
18.40  18.61   18.82 

15.07  15.25  15.44 
16.97  17.17  17.37 
19.03  19.25  19.47 

19 
21 
21 

2.7    19.683  19.90  20.12  20.35 
2.S!  21.952|  22.19  22.43  22.67 
2.9;  24.3S9|  24.64  24.90  25.15 

20.57  20.80  21.02 
22.91  23.15  23.39 
25.41  25.67  25.93 

21.25  21.48  21.72 
23.64  23.89  24.14 
26.20  26.46  26.73 

23 

25 
27 

3.0    27.000 

27.27  27.54  27.82 

28.09  28.37  28.65 

28.93  29.22  29.50 

29 

3.1!  29.791 
3.2   32.768 
3.3j  35.937 

30.08  30.37  30.66 
33.08  33.39  33.70 
36.26  36.59  36.93 

30.96  31.26  31.55 
34.01  34.33  34.65 
37.26  37.60  37.93 

31.86  32.16  32.46 
34.97  35.29  35.61 
38.27  38.61  38.96 

31 
33 
34 

3.4!  39.304 
3.5,  42.875 
3.6    46.656 

39.65  40.00  40.35 
43.24  43.61  43.99 
47.05  47.44  47.83 

40.71  41.06  41.42 
44.36  44.74  45.12 
48.23  48.63  49.03 

41.78  42.14  42.51 
45.50  45.88  46.27 
49.43  49.84  50.24 

37 
39 
41 

3.7  50.653'  51.06  51.48  51.90 

3.8  54.872  55.31  55.74  56.18 

3.9  59.319  59.78  60.24  60.70 

52.31  52.73  53.16 
56.62  57.07  57.51 
61.16  61.63  62.10 

53.58  54.01  54.44 
57.96  58.41  58.86 
62.57  63.04  63.52 

43 
46 

48 

4.0    64.000  64.48  64.96  65.45 

65.94  66.43  66.92 

67.42  67.92  68.42 

50 

4.1    68.921   69.43  69.93  70.44 
4.2,  74.088   74,62  75.15  75.69 
4.3    79.507  80.06  80.62  81.18 

70.96  71.47  71.99 
76.23  76.77  77.31 
81.75  82.31  82.88 

72.51  73.03  73.56 
77.85  78.40  78.95 
83.45  84.03  84.60 

53 
56 

58 

4.4    85.1841  85.77  86.35  86.94 
4..>    91.125'  91.73  92.35  92.96 
4.6   97.336,  97.97  98.61  99.25 

87.53  88.12  88.72 
93.58  94.20  94.82 
99.90  100.5  101.2 

89.31  89.92  90.52 
95.44  96.07  96.70 
101.8  102.5   103.2 

61 

64 

6 

4.7 

4.S 
4.9 

103.823,  104.5  105.2  105.8 
110.592!  111.3  112.0  112.7 
117.6^9  118.4  119.1  119.8 

106.5  107.2  107.9 
113.4  114,1  114.8 

120.6  121.3  122.0 

108.5  109.2  109.9 
115.5  116.2  116.9 
122.8  123.5  124.3 

7 
7 

7 

5.0 

125.000;  125.8  126.5  127.3 

128.0  128.8  129.6 

130.3  131.1  131.9 

8 

5.1  132.651S  133.4  134.2  135.0 

5.2  140.608;  141.4  142.2  143.1 

5.3  148.877   149.7  150.6  151.4 

135.8  136.6  137.4 

143.9  144.7  145.5 
152.3  153.1  154.0 

138.2  139.0  139.8 
146.4  147.2  148.0 
154.9  155.7  156.6 

8 
9 
9 

5.4  157.464    158.3  159.2  160.1 

5.5  I66.375:  167.3  168.2  169.1 

5.6  175.616   176.6  177.5  178.5 

161.0  161.9  162.8 
170.0  171.0  171.9 
179.4  180.4  181.3 

163.7  164.6  165.5 

172.8  173.7  174.7 
182.3  183.3  184.2 

9 
9 
10 

5.7  185.193   1S6.2  187.1  188.1 
5.S  195.n2:  196.1   197.1  198.2 
5.9  205.379  206.4  207.5  208.5 

189.1  190.1  191.1 

199.2  200.2  201.2 
209.6  210.6  211.7 

192.1  193.1   194.1 
202.3  203.3  204.3 
212.8  213.8  214.9 

10 
11 
11 

XV.     CUBES. 


39 


n 

0 

1         2        3 

4        5       6 

7         8        9 

d 

6.( 

)  216.000 

217.1  218.2  219.3 

220.3  221.4  222.5 

223.6  224.8  225.9 

11 

6.] 
6.1 
6.9 

L  226.981 

5  238.328 

250.047 

228.1  229.2  230.3 
239.5  240.6  241.8 

251.2  252.4  253.6 

231.5  232.6  233.7 
243.0  244.1  245.3 
254.8  256.0  257.3 

234.9  236.0  237.2 
246.5  247.7  248.9 
258.5  259.7  260.9 

11 
11 
12 

6.4 
6.a 
6.1 

[  262.144 

274.625 

»  287.496 

263.4  264.6  265.8 
275.9  277.2  278.4 

285.5  290.1  291.4 

267.1  268.3  269.6 

279.7  281.0  282.3 

292.8  294.1  295.4 

270.8  272.1  273.4 

283.6  284.9  286.2 

296.7  298.1  299.4 

12 
13 
14 

6.: 

6.S 

300.763 
314.432 
328.509 

302.1  303.5  304.8 

315.8  317.2  318.6 

329.9  331.4  332.8 

306.2  307.5  30S.9 
320.0  321.4  322.8 

334.3  335.7  337.2 

310.3  311.7  313.0 
324.2  325.7  327.1 
338.6  340.1  341.5 

14 
14 
15 

7.C 

343.000  344.5  345.9  347.4 

348.9  350.4  3il.9 

353.4  354.9  356.4 

15 

7.1 

7.2 
7.3 

357.911 
373.248 
389.017 

359.4  360.9  362.5 
374.8  376.4  377.9 
390.6  392.2  393.8 

364.0  365.5  367.1 
379.5  381.1  382.7 
395.4  397.1  398.7 

368.6  370.1  371.7 

384.2  385.8  387.4 

400.3  401.9  403.6 

IS 
16 
16 

7.4 

7.5 
7.G 

405.224 
421.875 
438.976 

406.9  408.5  410.2 

423.6  425.3  427.0 

440.7  442.5  444.2 

411.8  413.5  415.2 
428.7  430.4  432.1 

445.9  447.7  449.5 

416,8  418.5  420.2 
433.8  435.5  437.2 
451.2  453.0  454.8 

17 
18 
17 

7.7 

7.8 
7.9 

456.533 
■474.552 
493.039 

458.3  460.1  461.9 

476.4  478.2  480.0 
494.9  496.8  498.7 

463.7  465.5  467.3 
481.9  483.7  485.6 
500.6  502.5  504.4 

469.1  470.9  472.7 
487.4  489.3  491.2 
506.3  508.2  510.1 

19 
IS 
19 

8.C 

512.000 

513.9  515.8  517.8 

519.7  521.7  523.6 

525.6  527.5  529.5 

19 

8.1 

8.2 
8.3 

531.441 
551.368 
571.787 

533.4  535.4  537.4 
553.4  555.4  557.4 
573.9  575.9  578.0 

539.4  541.3  543.3 

559.5  561.5  563.6 
580.1  582.2  584.3 

545.3  547.3  .549.4 
565.6  567.7  569.7 

586.4  588.5  590.6 

20 
21 
21 

8.4 
8.5 
8.fc 

592.704 
614.125 
636.056 

594.8  596.9  599.1 
616.3  618.5  620.7 
638.3  6H0.5  642.7 

601.2  603.4  605.5 
622.8  625.0  627.2 
645.0  647.2  649.5 

607.6  609.8  612.0 
629.4  631.6  633.8 

651.7  654.0  656.2 

21 

23 
23 

8.7 
8.8 
8.9 

658.503 
681.472 
704.969 

660.8  663.1  665.3 
683.8  686.1  6885 
707.3  709.7  712.1 

667.6  669.9  672.2 
690.8  693.2  695.5 
714.5  716.9  719.3 

674.5  676.8  679.2 
697.9  700.2  702.6 
721.7  724.2  726.6 

23 
24 
24 

9.0 

729.000 

731.4  733.9  736.3 

738.8  741.2  743.7    746.1  748.6  751.1 

25 

9.1 
9.2 
9.3 

753.571 
778.688 
804.357 

756.1  758.6  761.0 

781.2  783.8  786.3 
807.0  809.6  812.2 

763.6  766.1  768.6    771.1  773.6  776.2 
788.9  791.5  794.0    796.6  799.2  801.8 
814.8  817.4  820.0  !  822.7  825.3  827.9 

25 
26 
27 

9.4  830.584 

9.5  857.375 

9.6  884.736 

833.2  835.9  838.6 
860.1  862.8  865.5 
887.5  890.3  893.1 

841.2  843.9  846.6    849.3  852.0  854.7 

868.3  871.0  873.7  ;  876.5  879.2  882.0 
895.8  898  6  901.4  :  904.2  907.0  909.9 

27 
27 
28 

9.7  912.673 

9.8  941.192 
9.9,970.299 

915.5  918.3  921.2 

944.1  947.0  949.9 

973.2  976.2  979.1 

924.0  926.9  929.7 
952.8  955.7  958.6 

982.1  985.1  988.0 

932.6  935.4  938.3 
961.5  964.4  967.4 
991.0  994.0  997.0 

29 
29 
30 

n 
T 

9 

3 

.0 

.1       .2      .3 

.4      .5      .6 

.7       .8      .9 

d 

1000 
0.125 
0.0  370 

*757  *579  *455 
108  *939  *822 
336    305     278 

*364  *296  *244 

*723  *640  *569 

254     233     214 

*204  *171  *146 

*.508  *456  *410 

197     182     169 

21 
40 
13 

4 
5 
6 

0.0  156 
0.00  800 
0.00  463 

145     135     126 
754     711     672 
441     420    400 

117     110     103 
635     601     569 
381     364    348 

*963  *904  *850 
540    513     487 
332     318    304 

50 

24- 

12 

7 
8 
9 

0.00  292 
0.00  195 
0.00  137 

279     268     257 
188     181     175 
133     129     124 

247     237     228 
169     163     157 
120     117     113 

219     211     203 
152     147     142 
110     106     103 

8 
5 
3 

40 


XVI.     SQUAEE  KOOTS 


V/i 


n 

0 

12   3 

4   5   6 

7    8   9 

d 

1.0 

1.0000 

0050  0100  0149 

0198  0247  0296 

0344  0392  0440 

48 

1.1 
1.2 
1.3 

04SS 
0954 
1402 

0536  0583  0630 
1000  1045  1091 
1446  1489  1533 

0677  0724  0770 
1136  1180  1225 
1576  1619  1662 

0817  0863  0909 
1269  1314  1358 
1705  1747  1790 

45 
44 

42 

1.4 
1.5 
1.0 

1S32 
2247 
2649 

1874  1916  1958 
2288  2329  2369 
2689  2728  2767 

2000  2042  2083 
2410  2450  2490 
2806  2845  2884 

2124  2166  2207 
2530  2570  2610 
2923  2961  3000 

40 
39 
38 

1.7 
1.8 
1.9 

303S 
3416 
37S4 

3077  3  Hi  3153 
3454  3491  3528 
3820  3856  3892 

3191  3229  3266 
3565  3601  3638 
3928  3964  4000 

3304  3342  3379 
3675  3711  3748 
4036  4071  4107 

37 
36 
35 

2.0 

1.4142 

4177  4213  4248 

4283  4318  4353 

4387  4422  4457 

34 

2.1 
2.2 
2.3 

4491 
4832 
5166 

4526  4560  4595 
4866  4900  4933 
5199  5232  5264 

4629  4663  4697 
4967  5000  5033 
5297  5330  5362 

4731  4765  4799 
5067  5100  5133 
5395  5427  5460 

33 
33 
32 

2.4 
2.5 
2.6 

5492 
5811 
6125 

5524  5556  5588 
5843  5875  5906 
6155  6186  6217 

5620  5652  5684 
5937  5969  6000 
6248  6279  6310 

5716  5748  5780 
6031  6062  6093 
6340  6371  6401 

31 
32 
31 

2.7 

2.8 
2.9 

6432 
6733 
7029 

6462  6492  6523 
6763  6793  6823 
7059  7088  7117 

6553  6583  6613 
6852  6882  6912 
7146  7176  7205 

6643  6673  6703 
6941  6971  7000 
7234  7263  7292 

30 
29 
29 

3.0 

1.7321 

7349  7378  7407 

7436  7464  7493 

7521  7550  7578 

29 

3.1 

3.2 
3.3 

7607 
7889 
8166 

7635  7664  7692 
7916  7944  7972 
8193  8221  8248 

7720  7748  7776 
8000  8028  8055 
8276  8303  8330 

7804  7833  7861 
8083  8111  8138 
8358  8385  8412 

28 
28 
27 

3.4 
3.5 
3.6 

8439 
8708 
8974 

S466  8493  8520 
8735  8762  8788 
9000  9026  9053 

8547  8574  8601 
8815  8841  8868 
9079  9105  9131 

8628  8655  8682 
8894  8921  8947 
9157  9183  9209 

26 
27 
26 

3.7 
3.8 
3.9 

9235 
9494 
9748 

9261  9287  9313 
9519  9545  9570 
9774  9799  9824 

9339  9365  9391 
9596  9621  9647 
9849  9875  9900 

9416  9442  9468 
9672  9698  9723 
9925  9950  9975 

26 
25 

25 

4.0 

2.0000 

0025  0050  0075 

0100  0125  0149 

0174  0199  0224 

24 

4.1 
4.2 
4.3 

0248 
0494 
0736 

0273  0298  0322 
0518  0543  0567 
0761  0785  0809 

0347  0372  0396 
0591  0616  0640 
0833  0857  0881 

0421  0445  0469 
0664  0688  0712 
0905  0928  0952 

25 
24 
24 

4.4 
4.5 
4.6 

0976 
1213 
1448 

1000  1024  1048 
1237  1260  1284 
1471  1494  1517 

1071  1095  1119 
1307  1331  1354 
1541  1564  1587 

1142  1166  1190 
1378  1401  1424 
1610  1633  1656 

23 
24 
23 

4.7 

4.8 
4.9 

1679 
1909 
2136 

1703  1726  1749 
1932  1954  1977 
2159  2181  2204 

1772  1794  1817 
2000  2023  2045 
2226  2249  2271 

1840  1863  1886 
2068  2091  2113 
2293  2316  2338 

23 
23 
23 

5.0 

2.2361 

2383  2405  2428 

2450  2472  2494 

2517  2539  2561 

22 

5.1 
5.2 
5.3 

2583 
2804 
3022 

2605  2627  2650 
2S25  2847  2869 
3043  3065  3087 

2672  2694  2716 
2891  2913  2935 
3108  3130  3152 

2738  2760  2782 
2956  2978  3000 
3173  3195  3216 

22 
22 
22 

5.4 
5.5 
5.6 

3238 
3452 
3664 

3259  3281  3302 
3473  3495  3516 
3685  3707  3728 

3324  3345  3367 
3537  3558  3580 
3749  3770  3791 

3388  3409  3431 
3601  3622  3643 
3812  3833  3854 

21 
21 
21 

5.7 

5.8 
5.9 

3875 
4083 
4290 

3896  3917  3937 
4104  4125  4145 
4310  4331  4352 

3958  3979  4000 
4166  4187  4207 
4372  4393  4413 

4021  4042  4062 
4228  4249  4269 
4434  4454  4474 

21 

21 
21 

V** 


riEST  PAET. 


41 


n 

0 

12   3 

4   5   6 

7    8   9 

d 

6.0 

2.4495 

4515  4536  4556 

4576  4597  4617 

4637  4658  4678 

20 

6.1 
6.2 
6.3 

4698 
4900 
5100 

4718  4739  4759 
4920  4940  4960 
5120  5140  5159 

4779  4799  4819 
4980  5000  5020 
5179  5199  5219 

4839  4860  4SS0 
5040  5060  5080 
5239  5259  5278 

20 
20 
20 

6.4 
6.5 
6.6 

5298 
5495 
5690 

5318  5338  5357 
5515  5534  5554 
5710  5729  5749 

5377  5397  5417 
5573  5593  5612 
5768  5788  5807 

5436  5456  5475 
5632  5652  5671 
5826  5846  5865 

20 
19 
19 

6.7 
6.8 
6.9 

5884 
6077 
6268 

5904  5923  5942 
6096  6115  6134 
6287  6306  6325 

5962  5981  6000 
6153  6173  6192 
6344  6363  6382 

6019  6038  6058 
6211  6230  6249 
6401  6420  6439 

19 
19 
19 

7.0 

2.6458 

6476  6495  6514 

6533  6552  6571 

6589  6608  6627 

19 

7.1 
7.2 
7.3 

6646 
6833 
7019 

6665  6683  6702 
6851  6870  6SS9 
7037  7055  7074 

6721  6739  6758 
6907  6926  6944 
7092  7111  7129 

6777  6796  6814 
6963  6981  7000 
7148  7166  7185 

19 
19 
18 

7.4 
7.5 
7.6 

7203 
7386 
7568 

7221  7240  7258 
7404  7423  7441 
7586  7604  7622 

7276  7295  7313 
7459  7477  7495 
7641  7659  7677 

7331  7350  7368 
7514  7532  7550 
7695  7713  7731 

18 
18 
18 

7.7 
7.8 
7.9 

7749 
7928 
8107 

7767  7785  7803 
7946  7964  7982 
8125  8142  8160 

7821  7839  7857 
8000  8018  8036 
8178  8196  8213 

7875  7893  7911 
8054  8071  8089 
8231  8249  8267 

17 
18 
17 

8.0 

2.8284 

8302  8320  8337 

8355  8373  8390 

8408  8425  8443 

17 

8.1 
8.2 
8.3 

8460 
8636 
8810 

8478  8496  8513 
8653  8671  8688 
8827  8844  8862 

8531  8548  8566 
8705  8723  8740 
8879  8896  8914 

8583  8601  8618 
8758  8775  8792 
8931  8948  8965 

18 
18 
IS 

8.4 
8.5 
8.6 

8983 
9155 
9326 

9000  9017  9034 
9172  9189  9206 
9343  9360  9377 

9052  9069  9086 
9223  9240  9257 
9394  9411  9428 

9103  9120  9138 
9275  9292  9309 
9445  9462  9479 

17 
17 
17 

8.7 
8.8 
8.9 

9496 
9665 
9833 

9513  9530  9547 
9682  9698  9715 
9850  9866  9883 

9563  9580  9597 
9732  9749  9766 
9900  9917  9933 

9614  9631  9648 
9783  9799  9816 
9950  9967  9983 

17 
17 
17 

"l6" 

9.0 

3.0000 

0017  0033  0050 

0067  0083  0100  0116  0133  0150 

9.1 
9.2 
9.3 

0166 
0332 
0496 

0183  0199  0216 
0348  0364  0381 
0512  0529  0545 

0232  0249  0265  0282  0299  0315 
0397  0414  0430  '  0447  0463  0480 
0561  0578  0594  0610  0627  0643 

17 
16 
16 

9.4 
9.5 
9.6 

0659 
0822 
0984 

0676  0692  0708 
0838  0854  0871 
1000  1016  1032 

0725  0741  0757  0773  0790  0806 
0887  0903  0919  0935  0952  0968 
1048  1064  1081  1097  1113  1129 

16 
16 
16 

9.7 
9.8 
9.9 

1145 
1305 
1464 

1161  1177  1193 
1321  1337  1353 
1480  1496  1512 

1209  1225  1241 
1369  13S5  1401 
1528  1544  1559 

1257  1273  1289 
1417  1432  1448 
1575  1591  1607 

16 
16 
16 

n 

.0 

.1   .2   .3 

.4  .5   .6 

.7   .8   .9 

d 

1 
2 
3 

1.000 
0.707 
0.577 

*953  *913  *877 
690  674  659 
568  559  550 

*845  *816  *791 
645  632  620 
542  535  527 

*767  *745  *725 
609  598  587 
520  513  506 

18 
10 
6 

4 
5 
6 

0.500 
0.447 
0.408 

494  488  482 
443  439  434 
405  402  398 

477  471  466 
430  426  423 
395  392  389 

461  456  452 
419  415  412 
386  383  381 

5 
4 
3 

7 
8 
9 

0.378 
0.354 
0.333 

375  373  370 
351  349  347 
331  330  328 

368  365  363 
345  343  341 
326  324  323 

360  358  356 
339  337  335 
321  319  318 

2 
2 

2 

42 


XVI.     SQUARE  EOOTS: 


Vn 


n 
10 

.0 

.1   .2  .3 

.4  .5  .6 

.7   .8  .9 

d 

3.1623 

1780  1937  2094 

2249  2404  2558 

2711  2863  3015 

151 

11 
J.J 

13 

3166 
4641 
6056 

3317  3466  3615 
4785  4928  5071 
6194  6332  6469 

3764  3912  4059 
5214  5355  5496 
6606  6742  6S7S 

4205  4351  4496 
5637  5777  5917 
7014  7148  7283 

145 
139 
134 

14 
15 
1« 

7417 

8730 

4.0000 

7550  7683  7815 
88.59  8987  9115 
0125  0249  0373 

7947  8079  8210 
9243  9370  9497 
0497  0620  0743 

8341  8471  8601 
9623  9749  9875 
0866  0988  1110 

129 
125 
121 

17 

18 
19 

1231 
2426 
3589 

1352  1473  1593 
2544  2661  2778 
3704  3818  3932 

1713  1833  1952 
2895  3012  3128 
4045  4159  4272 

2071  2190  2308 
3243  3359  3474 
4385  4497  4609 

118 
115 
112 

20 

4.4721 

4833  4944  5056 

5166  5277  5387 

5497  5607  5717 

109 

21 
23 

5826 
6904 
7958 

5935  6043  6152 
70li  7117  7223 
8062  8166  8270 

6260  6368  6476 
7329  7434  7539 
8374  8477  8580 

6583  6690  6797 
7645  7749  7854 
86S3  8785  S8SS 

107 
104 
102 

24 
25 
26 

8990 

5.0000 

0990 

9092  9193  9295 
0100  0200  0299 
1088  1186  1284 

9396  9497  9598 
0398  0498  0596 
1381  1478  1575 

9699  9800  9900 
0695  0794  0892 
1672  1769  1865 

100 
98 
97 

27 

28 
29 

1962 
2915 
3852 

2058  2154  2249 
3009  3104  3198 
3944  4037  4129 

2345  2440  2536 
3292  3385  3479 
4222  4314  4406 

2631  2726  2820 
3572  3666  3759 
4498  4589  4681 

95 
93 
91 

30 

5.4772 

4863  4955  5045 

5136  5227  5317 

5408  5498  55SS 

90 

31 
32 
33 

5678 
6569 
7446 

5767  5857  5946 
6657  6745  6833 
7533  7619  7706 

6036  6125  6214 
6921  7009  7096 
7793  7879  7966 

6303  6391  64 SO 
71S4  7271  7359 
8052  8138  8224 

89 
87 
86 

34 
35 
36 

8310 

9161 

6.0000 

8395  8481  8566 
9245  9330  9414 
0083  0166  0249 

8652  8737  8822 
9498  9582  9666 
0332  0415  0498 

8907  8992  9076 
9749  9833  9917 
0581  0663  0745 

85 
83 
S3 

37 

38 
39 

0828 
1644 
2450 

0910  0992  1074 
1725  1806  1887 
2530  2610  2690 

1156  1237  1319 
1968  2048  2129 
2769  2849  2929 

1400  1482  1563 
2209  2290  2370 
3008  3087  3166 

81 
80 
80 

40 

6.3246 

3325  3403  3482 

3561  3640  3718 

3797  3875  3953 

78 

41 
42 
43 

4031 
4807 

5574 

4109  4187  4265 
4885  4962  5038 
5651  5727  5803 

4343  4420  4498 
5115  5192  5269 
5879  5955  6030 

4576  4653  4730 
5345  5422  5498 
6106  6182  6257 

77 
76 

75 

44 
45 
46 

6332 
7082 
7823 

6408  6483  6558 
7157  7231  7305 
7897  7971  8044 

6633  6708  6783 
73SO  7454  7528 
8118  8191  8264 

6858  6933  7007 
7602  7676  7750 
8337  8411  8484 

75 
73 
73 

47 

48 
49 

8557 

9282 

7.0000 

8629  8702  8775 
9354  9426  9498 
0071  0143  0214 

8848  8920  8993 
9570  9642  9714 
0285  0356  0427 

9065  9138  9210 
9785  9857  9929 
0498  0569  0640 

72 

11 

70 

50 

7.0711 

0781  0852  0922 

0993  1063  1134 

1204  1274  1344 

51 
52 
53 

1414 
2111 
2801 

1484  1554  1624 
2180  2250  2319 
2870  2938  3007 

1694  1764  1833 
2388  2457  2526 
3075  3144  3212 

1903  1972  2042 
2595  2664  2732 
3280  3348  3417 

69 
69 
68 

64 
55 
56 

3485 
4162 
4833 

3553  3621  3689 
4229  4297  4364 
4900  4967  5033 

3756  3824  3892 
4431  4498  4565 
5100  5166  5233 

3959  4027  4095 
4632  4699  4766 
5299  5366  5432 

67 
67 
66 

57 

58 
59 

5498 
6158 
6811 

5565  5631  5697 
6223  6289  6354 
6877  6942  7006 

5763  5829  5895 
6420  6485  6551 
7071  7136  7201 

5961  6026  6092 
6616  6681  6746 
7266  7330  7395 

66 
65 

65 

Vn 


SECOND   PAET. 


43 


n 

.0 

.1   .2  .3 

.4   .5  .6 

.7   .8   .9 

(1 

60 

7.7460 

7524  7589  7653 

7717  7782  7846 

7910  7974  8038 

64 

61 
62 
63 

8102 
8740 
9373 

8166  8230  8294 
SS04  8867  8930 
9436  9498  9561 

8358  8422  8486 
8994  9057  9120 
9624  9687  9750 

8549  8613  8677 
91S3  9246  9310 
9812  9875  9937 

63 
63 
63 

64 
65 
66 

8.0000 
0623 
1240 

0062  0125  0187 
0685  0747  0S08 
1302  1363  1425 

0250  0312  0374 
0870  0932  0994 
1486  1548  1609 

0436  0498  0561 
1056  1117  1179 
1670  1731  1792 

62 
61 
62 

67 
68 
69 

1854 
2462 
3066 

1915  1976  2037 
2523  2583  2644 
3126  3187  3247 

2098  2158  2219 
2704  2765  2825 
3307  3367  3427 

2280  2341  2401 
2885  2946  3006 
3487  3546  3606 

61 
60 
60 

70 

8.3666 

3726  3785  3845 

3905  3964  4024 

4083  4143  4202 

59 

71 
72 
73 

4261 
4853 
5440 

4321  4380  4439 
4912  4971  5029 
5499  5557  5615 

4499  4558  4617 
5088  5147  5206 
5674  5732  5790 

4676  4735  4794 
5264  5323  5381 
5849  5907  5965 

59 
59 
58 

74 
75 
76 

6023 
6603 
7178 

6081  6139  6197 
6660  6718  6776 
7235  7293  7350 

6255  6313  6371 
6833  6891  6948 
7407  7464  7521 

6429  6487  6545 
7006  7063  7121 
7579  7636  7693 

58 
57 

57 

77 
78 
79 

7750 
8318 
8882 

7807  7864  7920 
8374  8431  8487 
8938  8994  9051 

7977  8034  8091 
8544  8600  8657 
9107  9163  9219 

8148  8204  8261 
8713  8769  8826 
9275  9331  9387 

57 
56 
56 

80 

8.9443 

9499  9554  9610 

9666  9722  9778 

9833  9889  9944 

56 

81 
82 
83 

9.0000 
0554 
1104 

0056  0111  0167 
0609  0664  0719 
1159  1214  1269 

0222  0277  0333 
0774  0830  0885 
1324  1378  1433 

0388  0443  0499 
0940  0995  1049 
148S  1542  1597 

55 
55 
55 

84 
85 
86 

1652 
2195 
2736 

1706  1761  1815 
2250  2304  2358 
2790  2844  2S98 

1869  1924  1978 
2412  2466  2520 
2952  3005  3059 

2033  2087  2141 
2574  2628  26S2 
3113  3167  3220 

54 
54 
54 

87 
88 
89 

3274 
3808 
4340 

3327  33S1  3434 
3862  3915  3968 
4393  4446  4499 

34SS  3541  3595 
4021  4074  4128 
4552  4604  4657 

3648  3702  3755 
4181  4234  4287 
4710  4763  4816 

53 
53 
52 

90 

9.4868 

4921  4974  5026 

5079  5131  5184 

5237  5289  5341 

53 

91 
92 
93 

5394 
5917 
6437 

5446  5499  5551 
5969  6021  6073 
6488  6540  6592 

5603  5656  5708 
6125  6177  6229 
6644  6695  6747 

5760  5812  5864 
6281  6333  6385 
6799  6850  6902 

53 

52 
52 

94 
95 

96 

6954 
7468 
7980 

7005  7057  7108 
7519  7570  7622 
8031  S0S2  8133 

7160  7211  7263 
7673  7724  7775 
8184  8234  S2S5 

7314  7365  7417 
7826  7877  7929 
S336  8387  S438 

51 
51 
51 

97 

98 
99 

8489 
8995 
9499 

8539  8590  8641 
9045  9096  9146 
9549  9599  9649 

S691  8742  8793 
9197  9247  9298 
9700  9750  9800 

8843  8894  8944 
9348  9398  9448 
9850  9900  9950 

51 
51 

50 

n 

0 

12   3 

4   5   6 

7    8   9 

d 

1 
2 
3 

0.316 
0.224 
0.183 

302  289  277 
219  213  209 
ISO  177  174 

267  258  250 
204  200  196 
171  169  167 

243  236  229 
192  189  186 
164  162  160 

5 
3 
2 

4 
5 
6 

0.158 
0.141 
0.129 

156  154  152 
140  139  137 
128  127  126 

151  149  147 
136  135  134 
125  124  123 

146  144  143 
132  131  130 
122  121  120 

2 

1 
0 

7 
8 
9 

0.120 
0.112 
0.105 

119  lis  117 
111  110  110 
105  104  104 

116  115  115 
109  108  108 
103  103  102 

114  113  113 
107  107  106 
102  101  101 

1 
1 
1 

XVII.     CUBE  ROOTS: 


</u 


n 

0 

12   3 

4   5   6 

7   8   9 

d 

1.0 

1.0000 

0033  0066  0099 

0132  0164  0196 

0228  0260  0291 

32 

1.1 
1.2 
1.3 

0323 
0627 
0914 

0354  0385  0416 
0656  06S5  0714 
0942  0970  0997 

0446  0477  0507 
0743  0772  0801 
1025  1052  1079 

0537  0567  0597 
0829  OS58  0886 
1106  1133  1160 

30 

28 
27 

1.4 
1.5 
l.« 

11S7 
1447 
1696 

1213  12-40  1266 
1473  1498  1523 
1720  174i  1769 

1292  1319  1344 
1.548  1573  1.598 
1793  1817  1840 

1370  1396  1422 
1623  1647  1672 
1S64  1888  1911 

25 
24 
24 

1.7 
l.S 
1.9 

193i 
2164 
2386 

1958  1981  2005 
2187  2209  2232 
2407  2429  2450 

2028  2051  2074 
2254  2276  2298 
2472  2493  2515 

2096  2119  2142 
2320  2342  2364 
2536  2557  2578 

22 
22 
21 

2.0 

1.2599 

2620  2641  2662 

2683  2703  2724 

2745  2765  2785 

21 

2.1 
2!3 

2S06 
3006 
3200 

2826  2846  2866 
3026  3045  3065 
3219  3238  3257 

2887  2907  2927 
3084  3104  3123 
3276  3295  3314 

2947  2966  2986 
3142  3162  3181 
3333  3351  3370 

20 
19 
19 

2.4 
2.5 
2.G 

3389 
3572 
3751 

3407  3426  3444 
3590  3608  3626 
3768  3786  3803 

3463  3481  3499 
3644  3662  3680 
3821  3838  3856 

3518  3536  3554 
3698  3715  3733 
3873  3890  3908 

18 
18 
17 

2.7 

2.S 
2.9 

3925 
4095 
4260 

3942  3959  3976 
4111  4128  4145 
4277  4293  4309 

3993  4010  4027 
4161  4178  4195 
4326  4342  4358 

4044  4061  4078 
4211  4228  4244 
4374  4390  4406 

17 
16 
16 

3.0 

1.4422 

4439  4454  4470 

4486  4502  4518 

4534  4550  4565 

16 

3.1 
3.2 
3.3 

4581 
4736 
4888 

4597  4612  4628 
4751  4767  4782 
4903  4918  4933 

4643  4659  4674 
4797  4812  4828 
4948  4963  4978 

4690  4705  4721 
4843  4858  4873 
4993  5007  5022 

15 
15 
15 

3.4 
3.5 
3.« 

5037 
5183 
5326 

5052  5066  5081 
5197  5212  5226 
5340  5355  5369 

5096  5110  5125 
5241  5255  5269 
5383  5397  5411 

5139  5154  5168 
5283  5298  5312 
5425  5439  5453 

15 
14 
14 

3.7 
3.8 
3.9 

5467 
5605 
5741 

5481  5495  5508 
5619  5632  5646 
5754  5767  5781 

5522  5536  5550 
5659  5673  5687 
5794  5808  5821 

5564  5577  5591 
5700  5714  5727 
5834  5848  5861 

14 
14 
13 

4.0 

1.5874 

5887  5900  5914 

5927  5940  5953 

5966  5979  5992 

13 

4.1 
4.2 
4.3 

6005 
6134 
6261 

6018  6031  6044 
6147  6160  6173 
6274  6287  6299 

6057  6070  6083 
6185  6198  6211 
6312  6324  6337 

6096  6109  6121 
6223  6236  6249 
6349  6362  6374 

13 
12 
12 

4.4 
4.5 
4.6 

6386 
6510 
6631 

6399  6411  6424 
6522  6534  6546 
6643  6655  6667 

6436  6448  6461 
6558  6571  6583 
6679  6691  6703 

6473  6485  6497 
6595  6607  6619 
6715  6727  6739 

13 
12 
12 

4.7 

4.8 
4.9 

6751 
6869 
6985 

6763  6774  6786 
6880  6892  6904 
6997  7008  7020 

6798  6810  6822 
6915  6927  6939 
7031  7043  7054 

6833  6S45  6857 
6950  6962  6973 
7065  7077  7088 

12 
12 
12 

5.0 

1.7100 

7111  7123  7134 

7145  7157  7168 

7179  7190  7202 

11 

5.1 
5.2 
5.3 

7213 
7325 
7435 

7224  7235  7247 
7336  7347  7358 
7446  7457  7468 

7258  7269  7280 
7369  7380  7391 
7479  7490  7501 

7291  7303  7314 
7402  7413  7424 
7512  7522  7533 

11 
11 
11 

5.4 
5.5 
5.6 

7544 
7652 

7758 

7555  7566  7577 
7662  7673  7684 
7769  7779  7790 

7587  7598  7609 
7694  7705  7716 
7800  7811  7821 

7620  7630  7641 
7726  7737  7748 
7832  7842  7853 

11 

10 
10 

5.7 

5.8 
5.9 

7863 
7967 
8070 

7874  7884  7894 
7977  7988  7998 
8080  8090  8100 

7905  7915  7926 
8008  8018  8029 
8110  8121  8131 

7936  7946  7957 
8039  8049  8059 
8141  8151  8161 

10 
11 
10 

-^n 


FIKST  PAKT. 


45 


n 

0 

12   3 

4   5   6 

7   8   9 

d 

6.0 
6.1 
6.2 
6.3 
6.4 
6.5 
6.6 
6.7 
6.8 
6.9 

1.8171 
8272 
8371 
8469 
8566 
8663 
8758 
8852 
8945 
9038 

8181  8191  8201 

8282  8292  8302 
8381  8391  8400 
8479  8489  8498 
8576  8586  8595 
8672  8682  8691 
8767  8777  8786 
8861  8871  8880 
8955  8964  8973 
9047  9056  9065 

8211  8222  8232 
8311  8321  8331 
8410  8420  8430 
8508  8518  8528 
8605  8615  8624 
8701  8710  8720 
8796  8805  8814 
8SS9  8899  8908 
8982  8992  9001 
9074  9084  9093 

8242  8252  8262 
8341  8351  8361 
8440  8450  8459 
8537  8547  8557 
8634  8643  8653 
8729  8739  8748 
8824  8833  8843 
8917  8927  8936 
9010  9019  9029 
9102  9111  9120 

10 
10 
10 
9 
10 
10 
9 
9 
9 
9 

7.0 
7.1 
7.2 
7.3 
7.4 
7.5 
7.6 
7.7 
7.8 
7.9 

1.9129 
9220 
9310 
9399 
9487 
9574 
9661 
9747 
9832 
9916 

9138  9148  9157 
9229  9238  9247 
9319  9328  9337 
9408  9416  9425 
9496  9504  9513 
9583  9592  9600 
9670  9678  9687 
9755  9764  9772 
9840  9849  9857 
9925  9933  9941 

9166  9175  9184 
9256  9265  9274 
9345  9354  9363 
9434  9443  9452 
9522  9531  9539 
9609  9618  9626 
9695  9704  9713 
9781  9789  9798 
9866  9874  9883 
9950  9958  9967 

9193  9202  9211 
9283  9292  9301 
9372  9381  9390 
9461  9469  9478 
9548  9557  9566 
9635  9644  9652 
9721  9730  9738 
9806  9815  9823 
9891  9899  9908 
9975  9983  9992 

9 
9 
9 
9 
8 
9 
9 
9 
8 
8 

8.0 
8.1 
8.2 
8.3 
8.4 
8.5 
8.6 
8.7 
8.8 
8.9 

2.0000 
0083 
0165 
0247 
0328 
0408 
0488 
0567 
0646 
0724 

0008  0017  0025 
0091  0100  0108 
0173  0182  0190 
0255  0263  0271 
0336  0344  0352 
0416  0424  0432 
0496  0504  0512 
0575  0583  0591 
0653  0661  0669 
0731  0739  0747 

0033  0042  0050 
0116  0124  0132 
0198  0206  0214 
0279  0288  0296 
0360  0368  0376 
0440  0448  0456 
0520  0528  0536 
0599  0606  0614 
0677  0685  0692 
0755  0762  0770 

0058  0066  0075 
0141  0149  0157 
0223  0231  0239 
0304  0312  0320 
0384  0392  0400 
0464  0472  0480 
0543  0551  0559 
0622  0630  0638 
0700  0708  0716 
0778  0785  0793 

8 
8 
8 
8 
8 
8 
8 
8 
8 
8 

9.0 
9.1 
9.2 
9.3 
9.4 
9.5 
9.6 
9.7 
9.8 
9.9 

2.0801 
0878 
0954 
1029 
1105 
1179 
1253 
1327 
1400 
1472 

0809  0816  0824 
0885  0893  0901 
0961  0969  0977 
1037  1045  1052 
1112  1120  1127 
1187  1194  1201 
1261  1268  1275 
1334  1341  1349 
1407  1414  1422 
1480  1487  1494 

0832  0839  0847 
0908  0916  0923 
0984  0992  0999 
1060  1067  1075 
1134  1142  1149 
1209  1216  1224 
1283  1290  1297 
1356  1363  1371 
1429  1436  1443 
1501  1508  1516 

0855  0862  0870 
0931  0939  0946 
1007  1014  1022 
1082  1090  1097 
1157  1164  1172 
1231  1238  1246 
1305  1312  1319 
1378  1385  1392 
1451  1458  1465 
1523  1530  1537 

8 
8 

7 
8 
7 
7 
8 
8 
7 
7 

n 

.0 

.1   .2  .3 

.4   .5   .6 

.7   .8  .9 

d 

I 

3 

t 

6 

7 
8 
9 

1.000 
0.794 
0.693 
0.630 
0.585 
0.550 
0.523 
0.500 
0.481 

*969  *941  *916 
781  769  758 
686  679  672 
625  620  615 
581  577  574 
547  544  541 
520  518  516 
498  496  494 
479  477  476 

*894  *874  *855 
747  737  727 
665  659  652 
610  606  601 
570  567  563 
539  536  533 
513  511  509 
492  490  488 
474  472  471 

*838  *822  *807 
718  709  701 
647  641  635 
597  593  589 
560  557  553 
530  528  525 
506  504  502 
486  484  483 
469  467  466 

13 

8 
5 
4 
3 
2 
2 
2 
2 

46 


XVII.     CUBE  EOOTSi 


</^ 


n 

.0 

.1  .2  .3 

.4  .6  .6 

.7  .8   .9 

d 

10 

2.1544 

1616  1687  1757 

1828  1898  1967 

2036  2104  2172 

68 

n 

12 
13 

2240 
2894 
3513 

2307  2374  2440 
2957  3021  3084 
3573  3633  3693 

2506  2572  2637 
3146  3208  3270 
3752  3811  3870 

2702  2766  2831 
3331  3.392  3453 
3928  3986  4044 

63 
60 

57 

14 
15 
IG 

4101 
4662 
5198 

41.59  4216  4272 
4717  4771  4825 
5251  5303  5355 

4329  4385  4441 
H879  4933  4987 
5407  5458  5509 

4497  4552  4607 
5040  5093  5146 
5561  5612  5662 

55 
52 
51 

17 

18 
19 

20~ 

5713 
6207 
6684 

2.7144 

5763  5813  5863 
6256  6304  6352 
6731  6777  6824 

5913  5962  6012 
6400  6448  6495 
6870  6916  6962 

6061  6110  6159 
6543  6590  6637 
7008  7053  7099 

48 
47 
45 

7189  7234  7279 

7324  7369  7413 

7457  7501  7545 

44 

21 
2^* 
23 

7589 
8020 
8439 

7633  7677  7720 
8063  8105  8147 
8480  8521  8562 

7763  7806  7850 
8189  8231  8273 
8603  8643  8684 

7S92  7935  7978 
8314  8356  8397 
8724  8765  8805 

42 
42 
40 

24 
25 
26 

8845 
9240 
9625 

8885  8925  8965 
9279  9318  9357 
9663  9701  9738 

9004  9044  9083 
9395  9434  9472 
9776  9814  9851 

9123  9162  9201 
9511  9549  9587 
9SS8  9926  9963 

39 
38 
37 

27 

28 
29 

3.0000 
0366 
0723 

0037  0074  0111 
0402  0438  0474 
0758  0794  0829 

0147  0184  0221 
0510  0546  0581 
0864  0899  0934 

0257  0293  0330 
0617  0652  0688 
0968  1003  1038 

36 
35 
34 

30 

3.1072 

1107  1141  1176 

1210  1244  1278 

1312  1346  1380 

34 

31 
32 
33 

1414 
1748 
2075 

1448  1481  1515 
1781  1814  1847 
2108  2140  2] 72 

1548  1582  1615 
1880  1913  1945 
2204  2237  2269 

1648  1682  1715 
1978  2010  2043 
2301  2332  2364 

33 
32 
32 

34 
35 
36 

2396 

2711 
3019 

2428  2460  2491 
2742  2773  2804 
3050  3080  3111 

2523  2554  2586 
2835  2866  2897 
314i  3171  3202 

2617  2648  2679 
2927  2958  2989 
3232  3262  3292 

32 
30 
30 

37 
38 

39 

3322 
3620 
3912 

3352  3382  3412 
3649  3679  3708 
3941  3970  3999 

3442  3472  3501 
3737  3767  3796 
4028  4056  4085 

3531  3561  3590 
3825  3854  3883 
4114  4142  4171 

30 
29 
29 

40 

3.4200 

4228  4256  4285 

4313  4341  4370 

4398  4426  4454 

28 

41 
42 
43 

4482 
4760 
5034 

4510  4538  4566 
4788  4815  4843 
5061  5088  5115 

4594  4622  4650 
4870  4898  4925 
5142  5169  5196 

4677  4705  4733 
4952  4980  5007 
5223  52.50  5277 

27 
27 
26 

44 
45 
46 

5303 
5569 
5830 

5330  5357  5384 
5595  5622  5648 
5856  5SS2  5908 

5410  5437  5463 
5674  5700  5726 
5934  5960  5986 

5490  5516  5543 
5752  5778  5805 
6011  6037  6063 

26 

25 
25 

47 

48 
49 

6088 
6342 
6593 

6114  6139  6165 
6368  6393  6418 
6618  6643  6668 

6190  6216  6241 
6443  6468  6493 
6692  6717  6742 

6267  6292  6317 
6518  6543  6568 
6766  6791  6816 

25 
25 
24 

50 

3.6840 

6S6i  6889  6914 

6938  6963  6987 

7011  7036  7060 

24 

61 
62 
53 

7084 
7325 
7563 

7109  7133  7157 
7349  7373  7397 
7586  7610  7634 

7181  7205  7229 
7421  7444  7468 
7657  7681  7704 

7253  7277  7301 
7492  7516  7539 
7728  7751  7774 

24 
24 
24 

54 
55 
56 

7798 
8030 
8259 

7821  7844  7868 
8053  8076  8099 
8281  8304  8327 

7891  7914  7937 
8121  8144  8167 
8349  8372  8395 

7960  7983  8006 
8190  8213  8236 
8417  8440  8462 

24 
23 
23 

57 

58 
59 

8485 
8709 
8930 

8508  8530  8552 
8731  8753  8775 
8952  8974  8996 

8575  8597  8620 
8798  8820  8842 
9018  90-10  9061 

8642  8664  8687 
8864  8886  8908 
9083  9105  9127 

22 
22 
22 

■Vn 


SECOND   PAET. 


47 


n 

.0 

.1   .2   .3 

.4  .5  .6 

.7  .8   .9 

d 

60 

3.9149 

9170  9192  9214 

9235  9257  9279 

9300  9322  9343 

22 

61 
62 
63 

9365 
9579 
9791 

9386  9408  9429 
9600  9621  9643 
9812  9833  9854 

9451  9472  9494 
9664  9685  9706 
9875  9896  9916 

9515  9536  9558 
9727  9748  9770 
9937  9958  9979 

21 
21 
21 

64 
65 
66 

4.0000 
0207 
(H12 

0021  0042  0062 
0228  0248  0269 
0433  0453  0474 

0083  0104  0125 
0290  0310  0331 
0494  0514  0534 

0145  0166  0187 
0351  0372  0392 
0555  0575  0595 

20 
20 
20 

67 

68 
69 

0615 
0817 
1016 

0636  0656  0676 
0837  0857  0877 
1035  1055  1075 

0696  0716  0736 
0896  0916  0936 
1095  1114  1134 

0756  0776  0797 
0956  0976  0996 
1154  1174  1193 

20 
20 
20 

70 

4.1213 

1232  1252  1272 

1291  1311  1330 

1350  1369  1389 

19 

71 
72 
73 

1408 
1602 
1793 

1428  1447  1466 
1621  1640  1659 
1812  1832  1851 

1486  1505  1524 
1679  1698  1717 
1870  1889  1908 

1544  1563  1582 
1736  1755  1774 
1927  1946  1964 

20 
19 
19 

74 
75 
76 

1983 
2172 
2358 

2002  2021  2040 
2190  2209  2228 
2377  2395  2414 

2059  2078  2097 
2246  2265  2284 
2432  2451  2469 

2115  2134  2153 
2302  2321  2340 
2488  2506  2525 

19 
18 
18 

77 
78 
79 

2543 

2727 
2908 

2562  2580  2598 
2745  2763  2781 
2927  2945  2963 

2617  2635  2653 
2799  2818  2836 
2981  2999  3017 

2672  2690  2708 
2854  2872  2890 
3035  3053  3071 

19 

18 
18 

80 

4.3089 

3107  3125  3143 

3160  3178  3196 

3214  3232  3250 

17 

81 
82 
83 

3267 
3445 
3621 

3285  3303  3321 
3463  34S0  3498 
3638  3656  3673 

3339  3356  3374 
3515  3533  3551 
3691  3708  3726 

3392  3409  3427 
3568  3586  3603 
3743  3760  3778 

18 
18 
17 

84 
85 
86 

3795 
3968 
4140 

3813  3830  3847 
3986  4003  4020 
4157  4174  4191 

3865  3882  3899 
4037  4054  4072 
4208  4225  4242 

3917  3934  3951 
4089  4106  4123 
4259  4276  4293 

17 
17 

17 

87 
88 
89 

4310 
4480 
4647 

4327  4344  4361 
4496  4513  4530 
4664  4681  4698 

4378  4395  4412 
4547  4564  4580 
4714  4731  4748 

4429  4446  4463 
4597  4614  4630 
4764  4781  4797 

17 
17 
17 

90 

4.4814 

4831  4847  4864 

4880  4897  4913 

4930  4946  4963 

16 

91 
92 
93 

4979 
5144 
5307 

4996  5012  5029 
5160  5176  5193 
5323  5339  5355 

5045  5062  5078 
5209  5225  5241 
5371  5388  5404 

5094  5111  5127 
5258  5274  5290 
5420  5436  5452 

17 
17 
16 

94 
95 
96 

5468 
5629 
5789 

5485  5501  5517 
5645  5661  5677 
5804  5820  5836 

5533  5549  5565 
5693  5709  5725 
5852  5868  5884 

5581  5597  5613 
5741  5757  5773 
5900  5915  5931 

16 
16 
16 

97 
98 
99 

5947 
6104 
6261 

5963  5979  5994 
6120  6136  6151 
6276  6292  6307 

6010  6026  6042 
6167  6183  6198 
6323  6338  6354 

6057  6073  6089 
6214  6229  6245 
6369  6385  6400 

15 
16 
16 

n 

0 

12   3 

4   5   6 

7   8   9 

d 

1 
2 
3 

0.464 
0.368 
0.322 

450  437  425 
362  357  352 
318  315  312 

415  405  397 
347  342  338 
309  306  303 

389  382  375 
333  329  325 
300  297  295 

7 
3 
3 

4 
5 
6 

0.292 
0.271 
0.255 

290  288  285 
270  268  266 
254  253  251 

283  281  279 
265  263  261 
250  249  247 

277  275  273 
260  258  257 
246  245  244 

2 
2 

1 

7 
8 
9 

0.243 
0.232 
0.223 

241  240  239 
231  230  229 

222  222  221 

238  237  236 
228  227  227 
220  219  218 

235  234  233 
226  225  224 
218  217  216 

1 
1 
1 

48 


XVII.     CUBE  EOOTS: 


■^ 


n 

0 

12   8 

4   5   6 

7   8   9 

d 

10 

4.6416 

6570  6723  6875 

7027  7177  7326 

7475  7622  7769 

145 

11 
12 

13 

7914 
9324 

5.065S 

8059  8203  8346 
9461  9597  9732 
0788  0916  1045 

8488  8629  8770 
9866*0000*0133 
1172  1299  1426 

8910  9049  9187 
♦0265*0397*0528 
1551  1676  1801 

137 
130 
124 

14 
15 
16 

192.^ 
3133 
42S8 

20^S  2171  2293 
3251  .3368  .3^S5 
4401  4514  4626 

2415  25.36  2656 
360i  3717  3832 
4737  4848  4959 

2776  2896  3015 
3947  4061  417.5 
5069  5178  5288 

118 
113 
109 

17 

18 
19 

5397 
6462 
7489 

5505  5613  5721 
6567  6671  6774 
7590  7690  7790 

5828  5934  6041 
6877  6980  7083 
7890  7989  8088 

6147  6252  6357 
7185  7287  7388 
8186  8285  8383 

105 
101 
97 

20 

5.8480 

8578  8675  8771 

8868  8964  9059 

9155  92.S0  9345 

94 

21 
23 

9439 

6.0368 

1269 

9533  9627  9721 
0459  0550  0641 
1358  1446  1534 

9814  9907*0000 
0732  0822  0912 
1622  1710  1797 

*0092*0185*0277 
1002  1091  1180 
1885  1972  2058 

91 

89 
87 

24 
25 
2G 

2145 
2996 

3825 

2231  2317  2403 
3080  3164  3247 
3907  39SS  4070 

2488  2573  2658 
3.3.30  3U3  .3496 
4151  4232  4312 

2743  2828  2912 
3579  3661  3743 
4393  4473  4553 

84 

82 
80 

27 

28 
29 

4633 
5421 
6191 

4713  4792  4872 
5499  5577  5654 
6267  6343  6419 

4951  .S030  5108 
5731  5808  5885 
6494  6569  6644 

5187  5265  5343 
5962  6039  6115 
6719  6794  6869 

78 
76 
74 

30 

6.6943 

7018  7092  7166 

7240  7313  7387 

7460  7533  7606 

73 

31 
32 
33 

7679 
8399 
9104 

7752  7824  7897 
8470  8541  8612 
9174  9244  9313 

7969  8041  8113 
8683  8753  8824 
9382  9451  9521 

8185  8256  8328 
8894  8964  9034 
9589  9658  9727 

71 
70 
68 

34 
35 
36 

9795 

7.0473 

1138 

9864  9932*0000 
0540  0607  Or.74 
1204  1269  1335 

*0068*0136*0203 
0740  0S07  0873 
1400  1466  1531 

*027 1*0338*0406 
0940  1006  1072 
1596  1661  1726 

67 
66 
65 

37 
38 
39 

1791 
2432 
3061 

1855  1920  1984 
2495  2558  2622 
3124  3186  3248 

2048  2112  2177 
2685  2748  2811 
3310  3372  3434 

2240  2304  2368 
2874  2936  2999 
3496  3558  3619 

64 
62 
62 

40 

7.3681 

3742  3803  3864 

3925  3986  4047 

4108  4169  4229 

61 

41 
42 
43 

4290 
4889 
5478 

4350  4410  4470 
4948  5007  5067 
5537  5595  .5654 

4530  4590  4650 
5126  51S5  5244 
5712  5770  5828 

4710  4770  4829 
5302  5361  5420 
5886  5944  6001 

60 

58 
58 

44 
45 
46 

6059 
6631 
7194 

6117  6174  6232 
6688  6744  6801 
7250  7306  7362 

6289  6346  6403 
6857  6914  6970 
7418  7473  7529 

6460  6517  6574 
7026  7082  7138 
7584  7639  7695 

57 
56 
55 

47 

48 
49 

7750 
8297 
8837 

7805  7860  7915 
8352  8-106  8460 
8891  8944  8998 

7970  8025  8079 
8514  8568  8622 
9051  9105  9158 

8134  8188  8243 
8676  8730  8784 
9211  9264  9317 

54 
53 
53 

50 

7.9370 

9423  9476  9528 

9581  9634  9686 

9739  9791  9843 

53 

51 
52 
53 

9896 

8.0415 

0927 

9948*0000*0052 
0466  0517  0569 
0978  1028  1079 

*01 04*01 56*0208 
0620  0671  0723 
1130  1180  1231 

*0260*03 11*0363 
0774  0825  0876 
1281  1332  1382 

52 
51 
51 

54 
55 
56 

1433 
1932 
2426 

1483  15.33  1583 
1982  2031  2081 
2475  2524  2573 

1633  1683  1733 
2130  2180  2229 
2621  2670  2719 

1783  1833  1882 
2278  2327  2377 
2768  2816  2865 

50 
49 
48 

57 

58 
59 

2913 
3396 

3872 

2962  3010  3059 
3443  3491  3539 
3919  3967  4014 

3107  3155  3203 
3587  3634  3682 
4061  4108  4155 

3251  3.300  3348 
3730  3777  3825 
4202  4249  4296 

.48 
47 
47 

Vn 


THIKD  PAET. 


49 


n 

0 

1   2   3 

4   5   6 

7   8   9 

d 

60 

8.4343 

4390  4437  4484 

4530  4577  4623 

4670  4716  4763 

46 

61 
62 
68 

4S09 
5270 
5726 

4856  4902  4948 
5316  5362  5408 
5772  5817  5862 

4994  5040  5086 
5453  5499  5544 
5907  5952  5997 

5132  5178  5224 
5590  5635  5681 
6043  6088  6132 

46 
45 
45 

64 
65 
66 

6177 
6624 
7066 

6222  6267  6312 
6668  6713  6757 
7110  7154  7198 

6357  6401  6446 
6801  6845  6890 
7241  7285  7329 

6490  6535  6579 
6934  6978  7022 
7373  7416  7460 

45 
44 
43 

67 

68 
69 

7503 
7937 
8366 

7547  7590  7634 
7980  8023  8066 
8408  8451  8493 

7677  7721  7764 
8109  8152  8194 
8536  8578  8621 

7807  7850  7893 
8237  8280  8323 
8663  8706  8748 

44 
43 
42 

70 

8.8790 

8833  8875  8917 

8959  9001  9043 

9085  9127  9169 

42 

71 
72 
73 

9211 

9628 

9.0041 

9253  9295  9337 
9670  9711  9752 
0082  0123  0164 

9378  9420  9462 
9794  9835  9876 
0205  0246  0287 

9503  9545  95S7 
9918  9959*0000 
0328  0369  0410 

41 
41 
40 

74 
75 
76 

0450 
0856 
1258 

0491  0532  0572 
0896  0937  0977 
1298  1338  1378 

0613  0654  0694 
1017  1057  1098 
1418  1458  1498 

0735  0775  0816 
1138  1178  1218 
1537  1577  1617 

40 
40 
40 

77 
78 
79 

1657 
2052 
2443 

1696  1736  1775 
2091  2130  2170 
2482  2521  2560 

1815  1855  1894 
2209  2248  2287 
2599  2638  2677 

1933  1973  2012 
2326  2365  2404 
2716  2754  2793 

40 
39 
39 

80 

9.2832 

2870  2909  2948 

2986  3025  3063 

3102  3140  3179 

38 

81 
82 
83 

3217 
3599 
3978 

3255  3294  3332 
3637  3675  3713 
4016  4053  4091 

3370  3408  3447 
3751  3789  3827 
4129  4166  4204 

3485  3523  3561 
3865  3902  3940 
4241  4279  4316 

38 
38 
38 

84 
85 
86 

4354 
4727 
5097 

4391  4429  4466 
4764  4801  4838 
5134  5171  5207 

4503  4541  4573 
4875  4912  4949 
5244  5281  5317 

4615  4652  4690 
4986  5023  5060 
5354  5391  5427 

37 
37 
37 

87 
88 
89 

5464 
5828 
6190 

5501  5537  5574 
5865  5901  5937 
6226  6262  6298 

5610  5647  5683 
5973  6010  6046 
6334  6370  6406 

5719  5756  5792 
6082  6118  6154 
6442  6477  6513 

36 
36 
36 

90 

9.6549 

6585  6620  6656 

6692  6727  6763 

6799  6834  6870 

35 

91 
92 
93 

6905 
7259 
7610 

6941  6976  7012 
7294  7329  7364 
7645  7680  7715 

7047  7082  7118 
7400  7435  7470 
7750  7785  7819 

7153  7188  7224 
7505  7540  7575 
7854  7889  7924 

35 
35 
35 

94 
95 
96 

7959 
8305 
8648 

7993  8028  8063 
8339  8374  8408 
8683  8717  8751 

8097  8132  8167 
8443  8477  8511 
8785  8819  8854 

8201  8236  8270 
8546  8580  8614 
8888  8922  8956 

35 
34 
34 

97 

98 
99 

8990 
9329 
9666 

9024  9058  9092 
9363  9396  9430 
9699  9733  9766 

9126  9160  9194 
9464  9497  9531 
9800  9833  9866 

9227  9261  9295 
9565  9598  9632 
9900  9933  9967 

34 
34 
33 

n 

0 

12   3 

4   5   6 

7   8   9 

d 

.1 
.2 
.3 

2.154 
1.710 
1.494 

087  027  *974 
682  657  632 
478  462  447 

*926  *882  *842 
609  587  567 
433  419  406 

*805  *771  *739 
547  529  511 
393  381  369 

29 
17 
12 

.4 
.5 
.6 

1.357 
1.260 
1.186 

346  335  325 
252  244  236 
179  173  167 

315  305  295 
228  221  213 
160  154  149 

286  277  268 
206  199  192 
143  137  132 

8 
6 
6 

.7- 
.8 
.9 

1.126 
1.077 
1.036 

121  116  111 
073  068  064 
032  028  025 

106  101  096 
060  056  052 
021  017  014 

091  086  082 
048  044  040 
010  007  003 

5 
4 
3 

50 


XVIII.     MULTIPLES. 


100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

1 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

2 

200 

202 

204 

206 

208 

210 

212 

214 

216 

218 

8 

300 

303 

306 

309 

312 

3r5 

318 

321 

324 

327 

4 

400 

404 

408 

412 

416 

420 

424 

428 

432 

436 

5 

500 

505 

510 

515 

520 

525 

530 

535 

540 

545 

(> 

600 

606 

612 

618 

624 

630 

636 

642 

648 

654 

7 

700 

707 

714 

721 

728 

735 

742 

749 

756 

763 

s 

8U0 

SOS 

816 

824 

832 

840 

848 

856 

864 

872 

9 

900 

909 

918 

927 

936 

945 

954 

963 

972 

981 

110 

111 

112 

113 

114 

115 

110 

117 

118 

119 

1 

110 

111 

112 

113 

114 

115 

116 

117 

118 

119 

2 

220 

222 

224 

226 

228 

230 

232 

234 

236 

238 

3 

330 

333 

336 

339 

342 

345 

348 

351 

354 

357 

4 

440 

444 

448 

452 

456 

460 

464 

468 

472 

476 

5 

550 

555 

560 

565 

570 

575 

580 

585 

590 

595 

6 

660 

666 

672 

678 

684 

690 

696 

702 

708 

714 

7 

770 

777 

784 

791 

798 

805 

812 

819 

826 

833 

8 

8S0 

SS8 

896 

904 

912 

920 

928 

936 

944 

952 

9 

990 

999  1008 

1017 

1026 

1035 

1044 

1053 

1062 

1071 

120 

121 

122 

123 

124 

125 

12G 

127 

128 

129 

1 

120 

121 

122 

123 

124 

"T25 

126 

~127 

128 

129 

2 

240 

242 

244 

246 

248 

250 

252 

254 

256 

258 

3 

360 

363 

366 

369 

372 

375 

378 

381 

3S4 

387 

4 

4S0 

484 

488 

492 

496 

500 

504 

508 

512 

516 

5 

600 

605 

610 

615 

620 

625 

630 

635 

640 

645 

6 

720 

726 

732 

738 

744 

750 

756 

762 

768 

774 

7 

840 

847 

854 

861 

868 

875 

882 

889 

896 

903 

8 

960 

968 

976 

984 

992 

1000  1008 

1016 

1024 

1032 

9 

lOSO 

1089 

1098 

1107 

1116 

1125 

1134 

1143 

1152 

1161 

130 

131 

132 

133 

134 

135 

136 

137 

138 

139 

1 

130 

131 

132 

133 

134 

135 

136 

137 

138 

139 

2 

260 

262 

264 

266 

268 

270 

272 

274 

276 

278 

3 

390 

393 

396 

399 

402 

405 

408 

411 

414 

417 

4 

520 

524 

528 

532 

536 

540 

544 

548 

552 

556 

5 

650 

655 

660 

665 

670 

675 

680 

685 

690 

695 

6 

780 

786 

792 

798 

804 

810 

816 

822 

828 

834 

7 

910 

917 

924 

931 

938 

945 

952 

959 

966 

973 

8 

1040 

1048 

1056 

1064 

1072 

lOSO 

10S8 

1096 

1104 

1112 

9 

1170 

1179 

IISS 

1197 

1206 

1215 

1224 

1233 

1242 

1251 

140 

141 

142 

143 

144 

145 

146 

147 

148 

149 

1 

140 

141 

142 

143 

144 

~145~ 

146 

147 

148 

149 

2 

280 

282 

284 

2S6 

288 

290 

292 

294 

296 

298 

3 

420 

423 

426 

429 

432 

435 

438 

441 

444 

447 

4 

560 

564 

568 

572 

576 

580 

584 

588 

592 

596 

5 

700 

705 

710 

715 

720 

725 

730 

735 

740 

745 

6 

840 

846 

852 

858 

864 

870 

876 

882 

888 

894 

7 

980 

987 

994 

1001 

lOOS 

1015 

1022 

1029 

1036 

1043 

8 

1120 

1128 

1136 

1144 

1152 

1160 

1168 

1176 

1184 

1192 

9 

1260 

1269 

1278 

1287 

1296 

1305 

1314 

1323 

1332 

1341 

XVIII.     MULTIPLES. 


51 


150 

151 

152 

153 

154  155 

156 

157 

158  159 

1 

150 

151 

152 

153 

154  155 

156 

157 

158  159 

2 

300 

302 

304 

306 

308  310 

312 

314 

316  318 

3 

450 

453 

456 

459 

462  465 

468 

471 

474  477 

4: 

600 

604 

60S 

612 

616  620 

624 

628 

632  636 

5 

750 

755 

760 

765 

770  775 

780 

785 

790  795 

6 

900 

906 

912 

918 

924  930 

936 

942 

948  954 

7 

1050 

1057 

1064 

1071 

1078  1085 

1092 

1099 

1106  1113 

8 

1200 

1208 

1216  1224 

1232  1240 

1248 

1256 

1264  1272 

9 

1350 

1359 

1368 

1377 

1386  1395 

1404 

1413 

1422  1431 

160 

161 

162 

163 

164  165 

166 

167 

168  169 

1 

160 

161 

162 

163 

164  165 

166 

167 

168  169 

2 

320 

322 

324 

326 

328  330 

332 

334 

336  338 

3 

480 

483 

486 

489 

492  495 

498 

501 

504  507 

4 

640 

644 

648 

652 

656  660 

664 

668 

672  676 

6 

800 

805 

810 

815 

820  825 

830 

835 

840  845 

6 

960 

966 

972 

978 

984  990 

996 

1002 

1008  1014 

7 

1120 

1127 

1134 

1141 

1148  1155 

1162 

1169 

1176  1183 

8 

1280 

1288 

1296 

1304 

1312  1320 

1328 

1336 

1344  1352 

9 

1440 

1449 

1458 

1467 

1476  1485 

1494 

1503 

1512  1521 

170 

171 

172 

173 

174  175 

176 

177 

178  179 

1 

170 

171 

172 

173 

174  175 

176 

177 

178  179 

2 

340 

342 

344 

346 

348  350 

352 

354 

356  358 

3 

510 

513 

516 

519 

522  525 

528 

531 

534  537 

4 

680 

684 

688 

692 

696  700 

704 

708 

712  716 

5 

850 

855 

860 

865 

870  875 

880 

SS5 

890  895 

6 

1020 

1026 

1032 

1038 

1044  1050 

1056 

1062 

1068  1074 

7 

1190 

1197 

1204 

1211 

1218  1225 

1232 

1239 

1246  1253 

8 

1360 

1368 

1376 

1384 

1392  1400 

1408 

1416 

1424  1432 

9 

1530 

1539 

1548 

1557 

1566  1575 

1584 

1593 

1602  1611 

180 

181 

182 

183 

184  185 

186 

187 

188  189 

1 

180 

181 

182 

183 

184  185 

186 

187 

188  189 

2 

360 

362 

364 

366 

368  370 

372 

374 

376  378 

3 

540 

543 

546 

549 

552  555 

558 

561 

564  567 

4 

720 

724 

728 

732 

736  740 

744 

748 

752  756 

5 

900 

905 

910 

915 

920  925 

930 

935 

940  945 

6 

1080 

1086 

1092 

1098 

1104  1110  1116 

1122 

1128  1134 

7 

1260 

1267 

1274 

1281 

1288  1295 

1302 

1309 

1316  1323 

8 

1440 

1448 

1456 

1464 

1472  14S0 

1488 

1496 

1504  1512 

9 

1620 

1629 

1638 

1647 

1656  1665 

1674 

1683 

1692  1701 

190 

191 

192 

193 

194  195 

196 

197 

198  199 

1 

190 

191 

192 

193 

194  195 

196 

197 

198  199 

2 

380 

382 

384 

386 

388  390 

392 

394 

396  398 

3 

570 

573 

576 

579 

582  585 

588 

591 

594  597 

4 

760 

764 

768 

772 

776  780 

784 

788 

792  796 

5 

950 

955 

960 

965 

970  975 

980 

985 

990  995 

6 

1140 

1146 

1152 

1158 

1164  1170 

1176 

1182 

1188  1194 

7 

1330 

1337 

1344 

1351 

1358  1365 

1372 

1379 

1386  1393 

8 

1520 

1528 

1536 

1544 

1552  1560 

1568 

1576 

1584  1592 

9 

1710 

1719 

1728 

1737 

1746  1755 

1764 

1773 

1782  1791 

52 


XVIII.     MULTIPLES. 


200 

201  202  203 

204 

205 

206 

207 

208  209 

1 

2 
3 

200 
400 
600 

201  202  203 
402  404  406 
603  606  609 

204 
408 
612 

205 
410 
615 

206 
412 
618 

207 
414 
621 

208  209 
416  418 
624  627 

4 
5 
G 

SOO 
1000 
1200 

804  808  812 
1005  1010  1015 
1206  1212  1218 

816 
1020 
1224 

820  824 
1025  1030 
1230  1236 

828 
1035 
1242 

832  836 
1040  1045 
1248  1254 

7 
8 
9 

1400 
1600 
ISOO 

1407  1414  1421 
1608  1616  1624 
1809  1818  1827 

1428 
1632 
1836 

1435 
1640 
1845 

1442 
1648 
1854 

1449 
1656 
1863 

1456  1463 
1664  1672 
1872  1881 

210 

211  212  213 

214 

215 

216 

217 

218  219 

1 
2 
3 

210 
420 
630 

211  212  213 
422  424  426 
633  636  639 

214 
428 
642 

215 
430 
645 

216 
432 
648 

217 
434 
651 

218  219 
436  438 
654  657 

4 
5 
0 

84  0 
1050 
1260 

844  848  852 
1055  1060  1065 
1266  1272  1278 

856 
1070 
1284 

860 
1075 
1290 

864 
1080 
1296 

868 
1085 
1302 

872  876 
1090  1095 
1308  1314 

7 
8 
9 

1470 
1680 
1890 

1477  1484  1491 
1688  1696  1704 
1899  1908  1917 

1498 
1712 
1926 

1505 
1720 
1935 

1512 
1728 
1944 

1519  1526  1533 
1736  1744  1752 
1953  1962  1971 

220 

221  222  223 

224 

225 

226 

227 

228  229 

1 
2 

3 

220 
440 
660 

221  222  223 
442  444  446 
663  666  669 

224 
448 
672 

225 
450 
675 

226 
452 
678 

227 
454 
681 

228  229 
456  458 
684  687 

4 
5 
6 

SSO 
1100 
1320 

884  888  892 
1105  1110  1115 
1326  1332  1338 

896  900  904 
1120  1125  1130 
1344  1350  1356 

908 
1135 
1362 

912  916 
1140  1145 
1368  1374 

7 
8 
9 

1540 
1760 
1980 

1547  1554  1561 
1768  1776  17S4 
1989  1998  2007 

1568  1575 
1792  1800 
2016  2025 

1582 
1808 
2034 

1589  1596  1603 
1816  1824  1832 
2043  2052  2061 

230 

231  232  233 

234 

235 

236 

237 

238  239 

1 
2 
3 

230 
460 
690 

231  232  233 
462  464  466 
693  696  699 

234 
468 
702 

235 
470 
705 

236 

472 
708 

237 
474 
711 

238  239 
476  478 
714  717 

4 
5 
6 

920 
1150 
1380 

924  928  932 
1155  1160  1165 
1386  1392  1398 

936 
1170 
1404 

940 
1175 
1410 

944 
1180 
1416 

948 
1185 
1422 

952  956 
1190  1195 
1428  1434 

7 
8 
9 

1610 
1840 
2070 

1617  1624  1631 
1848  1856  1864 
2079  2088  2097 

1638  1645 
1872  1880 
2106  2115 

1652 
1888 
2124 

1659 
1896 
2133 

1666  1673 
1904  1912 
2142  2151 

240 

241  242  243 

244 

245 

246 

247 

248  249 

1 
2 
3 

240 
480 
720 

241  242  243 
482  484  486 
723  726  729 

244 
488 
732 

245 
490 
735 

246 
492 
738 

247 
494 
741 

248  249 
496  498 
744  747 

4 
5 
6 

960 
1200 
1440 

964  968  972 
1205  1210  1215 
1446  1452  1458 

976 
1220 
1464 

980 
1225 
1470 

984 
1230 
1476 

988 
1235 
1482 

992  996 
1240  1245 
1488  1494 

7 
8 
9 

1680 
1920 
2160 

1687  1694  1701 
1928  1936  1944 
2169  2178  2187 

1708 
1952 
2196 

1715 
1960 

2205 

1722 
1968 
2214 

1729  1736  1743 
1976  1984  1992 
2223  2232  2241 

XVIII.     MULTIPLES. 


53 


250 

251  252  253 

254  255  256 

257  258  259 

1 

250 

251  252  253 

254  255  256 

257  258  259 

2 

500 

502  504  506 

508  510  512 

514  516  518 

3 

750 

753  756  759 

762  765  768 

771  774  777 

4 

1000 

1004  1008  1012 

1016  1020  1024 

1028  1032  1036 

5 

1250 

1255  1260  1265 

1270  1275  1280 

1285  1290  1295 

6 

1500 

1506  1512  1518 

1524  1530  1536 

1542  1548  1554 

7 

1750 

1757  1764  1771 

1778  1785  1792 

1799  1806  1813 

8 

2000 

2008  2016  2024 

2032  2040  2048 

2056  2064  2072 

9 

2250 

2259  2268  2277 

2286  2295  2304 

2313  2322  2331 

260 

261  262  263 

264  265  266 

267  268  269 

1 

260 

261  262  263 

264  265  266 

267  268  269 

2 

520 

522  524  526 

528  530  532 

534  536  538 

3 

780 

783  786  789 

792  795  798 

801  804  807 

4 

1040 

1044  1048  1052 

1056  1060  1064 

1068  1072  1076 

5 

1300 

1305  1310  1315 

1320  1325  1330 

1335  1340  1345 

6 

1560 

1566  1572  1578 

1584  1590  1596 

1602  1608  1614 

7 

1820 

1827  1834  1841 

1848  1855  1862 

1869  1876  1883 

8 

2080 

2088  2096  2104 

2112  2120  2128 

2136  2144  2152 

9 

2340 

2349  2358  2367 

2376  2385  2394 

2403  2412  2421 

270 

271  272  273 

274  275  276 

277  278  279 

1 

270 

271  272  273 

274  275  276 

277  278  279 

2 

540 

542  544  546 

548  550  552 

554  556  558 

3 

810 

813  816  819 

822  825  828 

831  834  837 

4 

1080 

1084  1088  1092 

1096  1100  1104 

1108  1112  1116 

5 

1350 

1355  1360  1365 

1370  1375  13S0 

1385  1390  1395 

6 

1620 

1626  1632  1638 

1644  1650  1656 

1662  1668  1674 

7 

1890 

1897  1904  1911 

1918  1925  1932 

1939  1946  1953 

8 

2160 

2168  2176  2184 

2192  2200  2208 

2216  2224  2232 

9 

2430 

2439  2448  2457 

2466  2475  2484 

2493  2502  2511 

280 

281  282  283 

284  285  286 

287  288  289 

1 

280 

281  282  283 

284  285  286 

287  288  289 

2 

560 

562  564  566 

568  570  572 

574  576  578 

3 

840 

843  846  849 

852  855  858 

861  864  867 

4 

1120 

1124  1128  1132 

1136  1140  1144 

1148  1152  1156 

5 

1400 

1405  1410  1415 

1420  1425  1430 

1435  1440  1445 

6 

1680 

1686  1692  1698 

1704  1710  1716 

1722  1728  1734 

7 

1960 

1967  1974  1981 

19SS  1995  2002 

2009  2016  2023 

8 

2240 

2248  2256  2264 

2272  2280  2288 

2296  2304  2312 

9 

2520 

2529  2538  2547 

2556  2565  2574 

2583  2592  2601 

290 

291  292  293 

294  295  296 

297  298  299 

1 

290 

291  292  293 

294  295  296 

297  298  299 

2 

580 

582  584  586 

588  590  592 

594  596  598 

3 

870 

873  876  879 

882  885  888 

891  894  897 

4 

1160 

1164  1168  1172 

1176  1180  1184 

1188  1192  1196 

5 

1450 

1455  1460  1465 

1470  1475  1480 

1485  1490  1495 

6 

1740 

1746  1752  1758 

1764  1770  1776 

1782  1788  1794 

7 

2030 

2037  2044  2051 

2058  2065  2072 

2079  2086  2093 

8 

2320 

2328  2336  2344 

2352  2360  2368 

2376  2384  2392 

9 

2610 

2619  2628  2637 

2646  2655  2664 

2673  26S2  2691 

XVIII.     MULTIPLES. 


800 

801 

802 

303 

304 

305 

306 

307 

308  309 

1 

2 
8 

300 
600 
900 

301 
602 
903 

302 
604 
906 

303 
606 
909 

304 
608 
912 

305 
610 
915 

306 
612 
918 

307 
614 
921 

308  309 
616  618 
924  927 

4 
5 
6 

1200 
1500 
ISOO 

1204 
1505 
1806 

1208 
1510 
1812 

1212 
1515 
1818 

1216 
1520 
1824 

1220 
1525 
1830 

1224 
1530 
1836 

1228 
1535 
1842 

1232  1236 
1540  1545 
1848  1854 

7 
H 
9 

2100 
2400 
2700 

2107 
2408 
2709 

2114 
2416 

2718 

2121 
2424 
2727 

2128 
2432 
2736 

2135 
2440 
2745 

2142 
2448 
2754 

2149 
2456 
2763 

2156  2163 
2464  2472 
2772  2781 

310 

311 

312 

313 

314 

315 

316 

317 

318  319 

1 
2 
3 

310 
620 
930 

311 
622 
933 

312 
624 
936 

313 
626 
939 

314 
628 
942 

315 
630 
945 

316 
632 
948 

317 
634 
951 

318  319 
636  638 
954  957 

4 
5 
6 

1240 
1550 
1860 

1244 
1555 
1866 

1248 
1560 
1872 

1252 
1565 
1878 

1256 
1570 
1884 

1260 
1575 
1890 

1264 
1580 
1896 

1268 
1585 
1902 

1272  1276 
1590  1595 
1908  1914 

7 
8 
9 

2170 
2480 
2790 

2177 
2488 
2799 

2184 
2496 
2808 

2191 
2504 
2817 

2198 
2512 
2826 

2205 
2520 
2835 

2212 
2528 
2844 

2219 
2536 

2853 

2226  2233 
2544  2552 
2862  2871 

320 

321 

322 

323 

324 

325 

326 

327 

328  329 

1 
2 
3 

320 
640 
960 

321 
642 
963 

322 
644 
966 

323 
646 
969 

324 
648 
972 

325 
650 
975 

326 

652 
978 

327 
654 
981 

328  329 
656  658 
984  987 

4 
5 

6 

1280 
1600 
1920 

1284 
1605 
1926 

1288 
1610 
1932 

1292 
1615 
1938 

1296 
1620 
1944 

1300 
1625 
1950 

1304 
1630 
1956 

1308 
1635 
1962 

1312  1316 
1640  1645 
1968  1974 

7 
8 
9 

2240 
2560 
2880 

2247 
2568 
2889 

2254 
2576 
2898 

2261 
2584 
2907 

2268 
2592 
2916 

2275 
2600 
2925 

2282 
2608 
2934 

2289 
2616 
2943 

2296  2303 
2624  2632 
2952  2961 

330 

331 

332 

333 

334 

335 

336 

337 

338  339 

1 
2 
3 

330 
660 
990 

331 
662 
993 

332 
664 
996 

333 
666 
999 

334 
668 
1002 

335 
670 

1005 

336 
672 
1008 

337 

674 
1011 

338  339 
676  678 
1014  1017 

4 
5 
6 

1320 
1650 
1980 

1324 
1655 
1986 

1328 
1660 
1992 

1332 
1665 
1998 

1336 
1670 
2004 

1340 
1675 
2010 

1344 
1680 
2016 

1348 
1685 
2022 

1352  1356 
1690  1695 
2028  2034 

7 

8 
9 

2310 
2640 
2970 

2317 
2648 
2979 

2324 
2656 

2988 

2331 
2664 
2997 

2338 
2672 
3006 

2345 
2680 
3015 

2352 
2688 
3024 

2359 
2696 
3033 

2366  2373 
2704  2712 
3042  3051 

340 

341 

342 

343 

344 

345 

346 

347 

348  349 

1 
2 
3 

340 
680 
1020 

341 
682 
1023 

342 
684 
1026 

343 

686 

1029 

344 
688 
1032 

345 
690 
1035 

346 
692 
1038 

347 

694 

1041 

348  349 
696  698 
1044  1047 

4 
5 
6 

1360 
1700 
2040 

1364 
1705 
2046 

1368 
1710 
2052 

1372 
1715 
2058 

1376 
1720 
2064 

1380 
1725 
2070 

1384 
1730 
2076 

1388 
1735 
2082 

1392  1396 
1740  1745 
2088  2094 

7 
8 
9 

2380 
2720 
3060 

2387 
2728 
3069 

2394 
2736 
3078 

2401 
2744 
3087 

2408 
2752 
3096 

2415 
2760 
3105 

2422 
2768 
3114 

2429 
2776 
3123 

2436  2443 
2784  2792 
3132  3141 

XVIII.     MULTIPLES. 


350 

351 

352 

353 

354  355 

356 

357 

358 

359 

1 

2 
3 

350 

700 

1050 

351 

702 

1053 

352 

704 

1056 

353 

706 

1059 

354  355 

70S   710 

1062  1065 

356 

712 

1068 

357 

714 

1071 

358 

716 

1074 

359 

718 

1077 

4 
5 
6 

1400 
1750 
2100 

1404 
1755 
2106 

1408 
1760 
2112 

1412 
1765 
2118 

1416  1420 
1770  1775 
2124  2130 

1424 
1780 
2136 

1428 
1785 
2142 

1432 
1790 
2148 

1436 
1795 
2154 

7 

8 
9 

2450 
2S00 
3150 

2457 
2808 
3159 

2464 
2816 
3168 

2471 
2824 
3177 

2478  2485 
2832  2840 
3186  3195 

2492 
2848 
3204 

2499 
2856 
3213 

2506 
2864 
3222 

2513 

2872 
3231 

360 

361 

362 

363 

364  365 

366 

367 

368 

369 

1 
2 
3 

360 

720 

1080 

361 
722 
1083 

362 
724 
1086 

363 
726 
1089 

364  365 
728  730 
1092  1095 

366 

732 

1098 

367 

734 

1101 

368 
736 
1104 

369 
738 
1107 

4 
5 

6 

1440 
1800 
2160 

1444 
1805 
2166 

1448 
1810 
2172 

1452 
1815 
2178 

1456  1460 
1820  1825 
2184  2190 

1464 
1830 
2196 

1468 
1835 
2202 

1472 
1840 
2208 

1476 
1845 
2214 

7 
8 
9 

2520 
2880 
3240 

2527 
2888 
3249 

2534 
2896 
3258 

2541 
2904 
3267 

2548  2555 
2912  2920 
3276  3285 

2562 
2928 
3294 

2569 
2936 
3303 

2576 
2944 
3312 

2583 
2952 
3321 

370 

371 

372 

373 

374  375 

376 

377 

378 

379 

1 
2 
3 

370 
740 
1110 

371 

742 

1113 

372 

744 

1116 

373 

746 

1119 

374  375 

748  750 

1122  1125 

376 

752 
1128 

377 

754 

1131 

378 
756 
1134 

379 

758 

1137 

4 
5 
6 

1480 
1850 
2220 

1484 
1855 
2226 

1488 
1860 
2232 

1492 
1865 
2238 

1496  1500 
1870  1875 
2244  2250 

1504 
1880 
2256 

1508 
1885 
2262 

1512 
1890 
2268 

1516 
1895 
2274 

7 
8 
9 

2590 
2960 
3330 

2597 
2968 
3339 

2604 
2976 
3348 

2611 
2984 
3357 

2618  2625 
2992  3000 
3366  3375 

2632 
3008 
3384 

2639 
3016 
3393 

2646 
3024 
3402 

2653 
3032 
3411 

380 

381 

382 

383 

384  385 

386 

387 

388 

389 

1 
2 

380 

760 

1140 

381 

762 

1143 

382 

764 

1146 

383 

766 

1149 

384  385 
768  770 
1152  1155 

386 

772 

1158 

387 
774 
1161 

388 

776 

1164 

389 

778 

1167 

4 
5 
6 

1520 
1900 
2280 

1524 
1905 
2286 

1528 
1910 

2292 

1532 
1915 
2298 

1536  1540 
1920  1925 
2304  2310 

1544 
1930 
2316 

1548 
1935 
2322 

1552 
1940 
2328 

1556 
1945 
2334 

7 
8 
9 

2660 
3040 
3420 

2667 
3048 
3429 

2674 
3056 
3438 

2681 
3064 
3447 

2688  2695 
3072  3080 
3456  3465 

2702 
3088 
3474 

2709 
3096 
3483 

2716 
3104 
3492 

2723 
3112 
3501 

390 

391 

392 

393 

394  395 

396 

397 

398 

399 

1 
2 
3 

390 
780 
1170 

391 

782 

1173 

392 

784 

1176 

393 

786 
1179 

394  395 

788   790 

1182  1185 

396 
792 
1188 

397 
794 
1191 

398 
796 
1194 

399 
798 
1197 

4 
5 
6 

1560 
1950 
2340 

1564 
1955 
2346 

1568 
1960 
2352 

1572 
1965 
2358 

1576  1580 
1970  1975 
2364  2370 

1584 
1980 
2376 

1588 
1985 
2382 

1592 
1990 
2388 

1596 
1995 
2394 

7 
8 
9 

2730 
3120 
3510 

2737 
3128 
3519 

2744 
3136 

3528 

2751 
3144 
3537 

2758  2765 
3152  3160 
3546  3555 

2772 
3168 
3564 

2779 
3176 
3573 

2786 
3184 
3582 

2793 
3192 
3591 

56 


XVIII.     MULTIPLES. 


400 

401 

402 

403 

404 

405 

406 

407 

408 

409 

1 

400 

401 

402 

403 

404 

405 

406 

407 

408 

409 

2 

SOO 

802 

804 

806 

SOS 

810 

812 

814 

816 

818 

3 

1200 

1203 

1206 

1209 

1212 

1215 

121S 

1221 

1224 

1227 

4 

1600 

1604 

1608 

1612 

1616 

1620 

1624 

1628 

1632 

1636 

5 

2000 

2005 

2010 

2015 

2020 

2025 

2030 

2035 

2040 

2045 

0 

2400 

2406 

2412 

2418 

2424 

2430 

2436 

2442 

2448 

2454 

7 

2S00 

2807 

2814 

2821 

2828 

2835 

2842 

2849 

2856 

2863 

8 

3200 

3208 

3216 

3224 

3232 

3240 

324B 

3256 

3264 

3272 

9 

3600 

3609 

3618 

3627 

3636 

3645 

3654 

3663 

3672 

3681 

410 

411 

412 

413 

414 

415 

416 

417 

418 

419 

1 

410 

411 

412 

413 

414 

415 

416 

417 

418 

419 

2 

S20 

822 

824 

826 

828 

830 

832 

834 

836 

838 

3 

1230 

1233 

1236 

1239 

1242 

1245 

1248 

1251 

1254 

1257 

4 

1640 

1644 

1648 

1652 

1656 

1660 

1664 

1668 

1672 

1676 

5 

2050 

2055 

2060 

2065 

2070 

2075 

2080 

2085 

2090 

2095 

6 

2460 

2466 

2472 

2478 

2484 

2490 

2496 

2502 

2508 

2514 

7 

2870 

2877 

2884 

2891 

2898 

2905 

2912 

2919 

2926 

2933 

8 

3280 

3288 

3296 

3304 

3312 

3320 

3328 

3336 

3344 

3352 

9 

3690 

3699 

3708 

3717 

3726 

3735 

3744 

3753 

3762 

3771 

420 

421 

422 

423 

424 

425 

426 

427 

428 

429 

1 

420 

421 

422 

423 

424 

425 

426 

427 

428 

429 

2 

840 

842 

844 

846 

848 

850 

852 

854 

856 

858 

3 

1260 

1263 

1266 

1269 

1272 

1275 

1278 

1281 

1284 

1287 

4 

1680 

1684 

1688 

1692 

1696 

1700 

1704 

1708 

1712 

1716 

5 

2100 

2105 

2110 

2115 

2120 

2125 

2130 

2135 

2140 

2145 

6 

2520 

2526 

2532 

2538 

2544 

2550 

2556 

2562 

2568 

2574 

7 

2940 

2947 

2954 

2961 

2968 

2975 

29S2 

2989 

2996 

3003 

8 

3360 

3368 

3376 

3384 

3392 

3400 

3408 

3416 

3424 

3432 

9 

3780 

3789 

3798 

3807 

3816 

3825 

3834 

3843 

3852 

3861 

430 

431 

432 

433 

434 

435 

436 

437 

438 

439 

1 

430 

431 

432 

433 

434 

435 

436 

437 

438 

439 

2 

860 

862 

864 

866 

868 

870 

872 

874 

876 

878 

3 

1290 

1293 

1296 

1299 

1302 

1305 

1308 

1311 

1314 

1317 

4 

1720 

1724 

1728 

1732 

1736 

1740 

1744 

1748 

1752 

1756 

5 

2150 

2155 

2160 

2165 

2170 

2175 

2180 

2185 

2190 

2195 

6 

2580 

2586 

2592 

2598 

2604 

2610 

2616 

2622 

2628 

2634 

7 

3010 

3017 

3024 

3031 

3038 

3045 

3052 

3059 

3066 

3073 

8 

3440 

3448 

3456 

3464 

3472 

3480 

3488 

3496 

3504 

3512 

9 

3870 

3879 

3SSS 

3897 

3906 

3915 

3924 

3933 

3942 

3951 

440 

441 

442 

443 

444 

445 

446 

447 

448 

449 

1 

440 

441 

442 

443 

444 

445 

446 

447 

44S 

449 

2 

880 

882 

884 

886 

888 

890 

892 

894 

896 

898 

3 

1320 

1323 

1326 

1329 

1332 

1335 

1338 

1341 

1344 

1347 

4 

1760 

1764 

1768 

1772 

1776 

1780 

1784 

1788 

1792 

1796 

5 

2200 

2205 

2210 

2215 

2220 

2225 

2230 

2235 

2240 

2245 

6 

2640 

2646 

2652 

2658 

2664 

2670 

2676 

2682 

2688 

2694 

7 

3080 

3087 

3094 

3101 

3108 

3115 

3122 

3129 

3136 

3143 

8 

3520 

3528 

3536 

3544 

3552 

3560 

3568 

3576 

3584 

3592 

9 

3960 

3969 

3978 

3987 

3996 

4005 

4014 

4023 

4032 

4041 

XVIII.     MULTIPLES. 


57 


450 

451 

452 

453 

454 

455 

456 

457 

458 

459 

1 

2 
3 

450 
900 
1350 

451 
902 
1353 

452 
904 
1356 

453 
906 
1359 

454 
908 
1362 

455 
910 
1365 

456 
912 
1368 

457 
914 
1371 

458 
916 
1374 

459 
918 
1377 

4 
5 
6 

ISOO 
2250 
2700 

1804 
2255 
2706 

1808 
2260 
2712 

1812 
2265 
2718 

1816 
2270 
2724 

1820 

2275 
2730 

1824 
2280 
2736 

1828 
2285 
2742 

1832 
2290 
2748 

1836 
2295 
2754 

7 
8 
9 

3150 
3600 
4050 

3157 
3608 
4059 

3164 
3616 
4068 

3171 
3624 

4077 

3178 
3632 
4086 

3185 
3640 
4095 

3192 
3648 
4104 

3199 
3656 
4113 

3206 
3664 
4122 

3213 
3672 
4131 

460 

461 

462 

463 

464 

465 

466 

467 

468 

469 

1 
2 
3 

460 
920 
1380 

461 
922 
1383 

462 
924 
1386 

463 
926 
1389 

464 
928 
1392 

465 
930 
1395 

466 
932 
1398 

467 
934 
1401 

468 
936 
1404 

469 
938 
1407 

4 
5 
6 

1840 
2300 
2760 

1844 
2305 
2766 

1848 
2310 
2772 

1852 
2315 
2778 

1856 
2320 
2784 

1S60 
2325 
2790 

1864 
2330 
2796 

1868 
2335 
2802 

1872 
2340 
2808 

1876 
2345 
2814 

7 

1 

3220 
3680 
4140 

3227 
3688 
4149 

3234 
3696 
4158 

3241 
3704 
4167 

3248 
3712 
4176 

3255 
3720 
4185 

3262 
3728 
4194 

3269 
3736 
4203 

3276 
3744 
4212 

3283 
3752 
4221 

470 

471 

472 

473 

iU 

475 

476 

477 

478 

479 

1 
2 
3 

470 
940 
1410 

471 
942 
1413 

472 
944 
1416 

473 
946 
1419 

474 
948 
1422 

475 
950 
1425 

476 
952 
1428 

477 
954 
1431 

478 
956 
1434 

479 
958 
1437 

4 
5 
6 

1880 
235a 
2820 

1884 
2355 
2826 

1888 
2360 
2832 

1892 
2365 
2838 

1896 
2370 
2844 

1900 
2375 
2850 

1904 
2380 
2856 

1908 

2385 
2862 

1912 
2390 
2868 

1916 
2395 
2874 

7 

8 
9 

3290 
3760 
4230 

3297 
3768 
4239 

3304 
3776 
4248 

3311 
3784 
4257 

3318 
3792 
4266 

3325 
3800 
4275 

3i32 

3808 
4284 

3339 
3816 
4293 

3346 
3824 
4302 

3353 
3832 
4311 

480 

481 

482 

483 

484 

485 

486 

487 

488 

489 

1 

2 

3 

480 
960 
1440 

481 
962 
1443 

482 
964 
1446 

483 
966 
1449 

484 
968 
1452 

485 
970 
1455 

486 
972 
1458 

487 
974 
1461 

488 
976 
1464 

489 
978 
1467 

4 
5 
6 

1920 
2400 
2S80 

1924 
2405 
2886 

1928 
2410 
2892 

1932 
2415 
2898 

1936 
2420 
2904 

1940 
2425 
2910 

1944 
2430 
2916 

1948 
2435 
2922 

1952 
2440 
2928 

1956 
2445 
2934 

7 
8 
9 

3360 
3840 
4320 

3367 
3848 
4329 

3374 
3856 
4338 

3381 
3864 
4347 

3388 
3872 
4356 

3395 
3880 
4365 

3402 
3888 
4374 

3409 
3896 
4383 

3416 
3904 
4392 

3423 
3912 
4401 

490 

491 

492 

493 

494 

495 

496 

497 

498 

499 

1 
2 
3 

490 
980 
1470 

491 
982 
1473 

492 
984 
1476 

493 
986 
1479 

494 
988 
1482 

495 
990 
1485 

496 
992 

1488 

497 
994 
1491 

498 
996 
1494 

499 
998 
1497 

4 
5 
6 

1960 
2450 
2940 

1964 
2455 
2946 

1968 
2460 
2952 

1972 
2465 
2958 

1976 
2470 
2964 

1980 
2475 
2970 

1984 
2480 
2976 

1988 
2485 
2982 

1992 
2490 
2988 

1996 
2495 
2994 

7 

8 
9 

3430 
3920 
4410 

3437 
3928 
4419 

3444 
3936 
4428 

3451 
3944 
4437 

3458 
3952 
4446 

3465 
3960 
4455 

3472 
3968 
4464 

3479 
3976 
4473 

3486 
3984 
4482 

3493 
3992 
4491 

58 


XVIII.     MULTIPLES. 


6(M> 

501 

502 

503 

504 

505 

506 

507 

608 

509 

1 

2 
8 

500 
1000 
1500 

501 
1002 
1503 

502 
1004 
1506 

503 
1006 
1509 

504 
1008 
1512 

505 
1010 
1515 

506 
1012 
1518 

507 
1014 
1521 

508 
1016 
1524 

509 
1018 
1527 

4 
5 

2000 
2500 
3000 

2004 
2505 
3006 

2008 
2510 
3012 

2012 
2515 
3018 

2016 
2520 
3024 

2020 
2525 
3030 

2024 
2530 
3036 

2028 
2535 
3042 

2032 
2540 
3048 

2036 
2545 
3054 

s 

9 

3500 
4000 
4500 

3507 
4008 
4509 

3514 
4016 
4518 

3521 
4024 
4527 

3528 
4032 
4536 

3535 
4040 
4545 

3542 
4048 
4554 

3549 
4056 
4563 

3556 
4064 

4572 

3563 
4072 

4581 

510 

511 

612 

513 

514 

515 

510 

517 

518 

519 

1 
2 
3 

510 
1020 
1530 

511 
1022 
1533 

512 
1024 
1536 

513 
1026 
1539 

514 
1028 
1542 

515 
1030 
1545 

516 
1032 
1548 

517 
1034 
1551 

518 
1036 
1554 

519 
1038 
1557 

4 
5 
6 

2(H0 
2550 
3060 

2044 
2555 
3066 

2048 
2560 
3072 

2052 
2565 
3078 

2056 
2570 
3084 

2060 
2575 
3090 

2064 
2580 
3096 

2068 
2585 
3102 

2072 
2590 
3108 

2076 
2595 
3114 

7 
8 
9 

3570 
4080 
4590 

3577 
4088 
4599 

3584 
4096 
4608 

3591 
4104 
4617 

3598 
4112 
4626 

3605 
4120 
4635 

3612 
4128 
4644 

3619 
4136 
4653 

3626 
4144 
4662 

3633 
4152 
4671 

520 

521 

522 

523 

524 

525 

526 

527 

528 

529 

1 
2 

3 

520 
1040 
1560 

521 
1042 
1563 

522 
1044 
1566 

523 
1046 
1569 

524 
1043 
1572 

525 
1050 
1575 

526 
1052 
1578 

527 
1054 
1581 

528 
1056 
1584 

529 
1058 
1587 

4 
5 

6 

20S0 
2600 
3120 

2084 
2605 
3126 

2088 
2610 
3132 

2092 
2615 
3138 

2096 
2620 
3144 

2100 
2625 
3150 

2104 
2630 
3156 

2108 
2635. 
3162 

2112 
2640 
3168 

2116 
2645 
3174 

7 
S 
9 

3640 
4160 
4680 

3647 
4168 
4689 

3654 
4176 
4698 

3661 
4184 

4707 

3668 
4192 
4716 

3675 
4200 
4725 

3682 
4208 
4734 

3689 
4216 
4743 

3696 
4224 

4752 

3703 
4232 
4761 

530 

531 

532 

533 

534 

535 

536 

537 

538 

539 

1 
2 
3 

530 
1060 
1590 

531 
1062 
1593 

532 
1064 
1596 

533 
1066 
1599 

534 
1068 
1602 

535 
1070 
1605 

536 
1072 
1608 

537 
1074 
1611 

538 
1076 
1614 

539 
1078 
1617 

4 
5 
6 

2120 
2650 
3180 

2124 
2655 
3186 

2128 
2660 
3192 

2132 
2665 
3198 

2136 
2670 
3204 

2140 
2675 
3210 

2144 
26S0 
3216 

2148 
2685 
3222 

2152 
2690 
3228 

2156 
2695 
3234 

7 

8 
9 

3710 
4240 
4770 

3717 
4248 
4779 

3724 
4256 

4788 

3731 
4264 
4797 

3738 
4272 
4806 

3745 
4280 

4815 

3752 
42SS 
4824 

3759 
4296 
4833 

3766 
4304 
4842 

3773 
4312 
4851 

540 

541 

542 

543 

544 

545 

546 

547 

548 

549 

1 
2 
3 

540 
1080 
1620 

541 
1082 
1623 

542 
1084 
1626 

543 
1086 
1629 

544 
1088 
1632 

545 
1090 
1635 

546 
1092 
1638 

547 
1094 
1641 

548 
1096 
1644 

549 
1098 
1647 

4 
a 
6 

2160 
2700 
3240 

2164 
2705 
3246 

216S 
2710 
3252 

2172 
2715 
3258 

2176 
2720 
3264 

2180 
2725 
3270 

2184 
2730 
3276 

2188 
2735 
32S2 

2192 
2740 
3288 

2196 

2745 
3294 

7 

8 
9 

3780 
4320 
4860 

3787 
4328 
4869 

3794 
4336 

4878 

3801 
4344 

48S7 

3808 
4352 
4896 

3815 
4360 
4905 

3822 
4368 
4914 

3829 
4376 
4923 

3836 
4384 
4932 

3843 
4392 
4941 

XVIII.    MULTIPLES. 


59 


550 

551 

552 

553 

554 

555 

556 

557 

558 

559 

1 

2 
3 

550 
1100 
1650 

551 
1102 
1653 

552 
1104 
1656 

553 
1106 
1659 

554 
1108 
1662 

555 
1110 
1665 

556 
1112 
1668 

557 
1114 
1671 

558 
1116 
1674 

559 
1118 
1677 

4 
5 
6 

2200 
2750 
3300 

2204 
2755 
3306 

2208 
2760 
3312 

2212 
2765 
3318 

2216 
2770 
3324 

2220 
2775 
3330 

2224 
2780 
3336 

2228 
2785 
3342 

2232 
2790 
3348 

2236 
2795 
3354 

7 
8 
9 

3S50 
4400 
4950 

3857 
4408 
4959 

3864 
4416 
4968 

3871 
4424 
4977 

3878 
4432 
4986 

3885 
4440 
4995 

3892 
4448 
5004 

3899 
4456 
5013 

3906 
4^64 
5022 

3913 
4472 
5031 

560 

561 

562 

563 

564 

565 

566 

567 

568 

569 

1 
2 
3 

560 
1120 
16S0 

561 
1122 
1683 

562 
1124 
1686 

563 
1126 
1689 

564 
1128 
1692 

565 
1130 
1695 

566 
1132 
1698 

567 
1134 
1701 

568 
1136 
1704 

569 
1138 
1707 

4 
5 

2240 
2800 
3360 

2244 
2805 
3366 

2248 
2810 
3372 

2252 
2815 
3378 

2256 
2820 
3384 

2260 
2825 
3390 

2264 
2830 
3396 

2268 
2835 
3402 

2272 
2840 
3408 

2276 
2845 
3414 

7 
8 
9 

3920 
4480 
5040 

3927 
4488 
5049 

3934 
4496 
5058 

3941 
4504 
5067 

3948 
4512 
5076 

3955 
4520 
5085 

3962 
4528 
5094 

3969 
4536 
5103 

3976 
4544 
5112 

3983 
4552 
5121 

570 

571 

572 

573 

574 

575 

576 

577 

578 

579 

1 
2 
3 

570 
1140 
1710 

571 
1142 
1713 

572 
1144 
1716 

573 
1146 
1719 

574 
1148 
1722 

575 
1150 
1725 

576 
1152 
1728 

577 
1154 
1731 

578 
1156 
1734 

579 
1158 
1737 

4 
5 

G 

2280 
2850 
3420 

2284 
2855 
3426 

2288 
2860 
3432 

2292 
2865 
3438 

2296 
2870 
3444 

2300 

2875 
3450 

2304 
2880 
3456 

2308 
2885 
3462 

2312 
2890 
3468 

2316 
2895 
3474 

7 
8 
9 

3990 
4560 
5130 

3997 
4568 
5139 

4004 
4576 

5148 

4011 
4584 

5157 

4018 
4592 
5166 

4025 
4600 

5175 

4032 
4608 
5184 

4039 
4616 
5193 

4046 
4624 
5202 

4053 
4632 
5211 

580 

581 

582 

583 

584 

585 

586 

587 

588 

589 

1 
2 
3 

580 
1160 
1740 

581 
1162 
1743 

582 
1164 
1746 

583 
1166 
1749 

584 
1168 
1752 

585 
1170 
1755 

586 
1172 
1758 

587 
1174 
1761 

588 
1176 
1764 

589 
1178 
1767 

4 
5 

6 

2320 
2900 
3480 

2324 
2905 
3486 

2328 
2910 
3492 

2332 
2915 
3498 

2336 
2920 
3504 

2340 
2925 
3510 

2344 
2930 
3516 

2348 
2935 
3522 

2352 
2940 
3528 

2356 
2945 
3534 

7 
8 
9 

4060 
4640 
5220 

4067 
4648 
5229 

4074 
4656 
5238 

4081 
4664 
52-17 

4088 
4672 
5256 

4095 
4680 
5265 

4102 
4688 
5274 

4109 
4696 

5283 

4116 
4704 
5292 

4123 
4712 
5301 

590 

591 

592 

593 

594 

595 

596 

597 

598 

599 

1 
2 
3 

590 
1180 
1770 

591 
1182 
1773 

592 
1184 
1776 

593 
1186 
1779 

594 
1188 
1782 

595 
1190 
1785 

596 
1192 
1788 

597 
1194 
1791 

598 
1196 
1794 

599 
1198 
1797 

4 
5 
6 

2360 
2950 
3540 

2364 
2955 
3546 

2368 
2960 
3552 

2372 
2965 
3558 

2376 
2970 
3564 

2380 
2975 
3570 

2384 
2980 
3576 

2388 
2985 
3582 

2392 
2990 
3588 

2396 
2995 
3594 

7 
8 
9 

4130 
4720 
5310 

4137 
4728 
5319 

4144 
4736 
5328 

4151 

4744 
5337 

4158 
4752 
5346 

4165 
4760 
5355 

4172 

4768 
5364 

4179 
4776 
5373 

4186 
4784 
5382 

4193 
4792 
5391 

60 


XVIII.     MULTIPLES. 


600 

601 

602 

603 

604 

605 

606 

607 

608  609 

1 

2 
8 

600 
1200 
1800 

601 
1202 
1803 

602 
1204 
1806 

603 
1206 
1809 

604 
1208 
1812 

605 
1210 
1815 

606 

1212 
1818 

607 
1214 
1821 

608  609 
1216  1218 
1824  1827 

4 
5 
6 

2400 
3000 
3600 

2404 
3005 
3606 

2408 
3010 
3612 

2412 
3015 
3618 

2416 
3020 
3624 

2420 
3025 
3630 

2424 
3030 
3636 

2428 
3035 
3642 

2432  2436 
3040  3045 
3648  3654 

7 

1 

4200 
4800 
5400 

4207 
4808 
5409 

4214 
4816 
5418 

4221 
4824 

5427 

4228 
4832 
5436 

4235 
4840 
5445 

4242 
4848 
5454 

4249 
4856 
5463 

4256  4263 
4864  4872 
5472  5481 

610 

611 

612 

613 

614 

615 

616 

617 

618  619 

1 
2 
3 

610 
1220 
1830 

611 
1222 
1833 

612 
1224 
1836 

613 
1226 
1839 

614 
1228 
1842 

615 
1230 
1845 

616 
1232 

1848 

617 
1234 
1851 

618  619 
1236  1238 
1854  1857 

4 
5 

2440 
3050 
3660 

2444 
3055 
3666 

2448 
3060 
3672 

2452 
3065 
3678 

2456 
3070 
3684 

2460 
3075 
3690 

2464 
3080 
3696 

2468 
3085 
3702 

2472  2476 
3090  3095 
3708  3714 

7 

8 
9 

4270 
4SS0 
5490 

4277 
4888 
5499 

4284 
4896 
5508 

4291 
4904 
5517 

4298 
4912 
5526 

4305 
4920 

5535 

4312 
4928 
5544 

4319 
4936 
5553 

4326  4333 
4944  4952 
5562  5571 

6-'0 

621 

622 

623 

624 

625 

626 

627 

628  629 

1 
2 
3 

620 
1240 
I860 

621 
1242 
1863 

622 
1244 
1866 

623 
1246 
1869 

624 
1248 
1872 

625 
1250 

1875 

626 
1252 
1878 

627 
1254 
1881 

628  629 
1256  1258 
1884  1887 

4 
5 
6 

2480 
3100 
3720 

2484 
3105 
3726 

2488 
3110 
3732 

2492 
3115 
3738 

2496 
3120 
3744 

2500 
3125 
3750 

2504 
3130 
3756 

2508 
3135 
3762 

2512  2516 
3140  3145 
3768  3774 

7 

8 
9 

4340 
4960 
5580 

4347 
4968 
5589 

4354 
4976 

5598 

4361 
4984 

5607 

4368 
4992 
5616 

4375 

5000 
5625 

4382 
5008 
5634 

4389 
5016 
5643 

4396  4403 
5024  5032 
5652  5661 

G30 

631 

632 

633 

634 

635 

636 

637 

638  639 

1 
2 
3 

630 
1260 
1890 

631 
1262 
1893 

632 
1264 
1896 

633 
1266 
1899 

634 
1268 
1902 

635 
1270 
1905 

636 

1272 
1908 

637 
1274 
1911 

638  639 
1276  1278 
1914  1917 

4 
5 

G 

2520 
3150 
3780 

2524 
3155 
3786 

2528 
3160 
3792 

2532 
3165 
3798 

2536 
3170 
3804 

2540 
3175 
3810 

2544 
3180 
3816 

2548 
3185 
3822 

2552  2556 
3190  3195 
3828  3834 

7 
8 
9 

4410 
5040 
5670 

4417 
5048 

5679 

4424 
5056 
5688 

4431 
5064 

5697 

4438 
5072 
5706 

4445 
5080 
5715 

4452 
5088 

5724 

4459 
5096 
5733 

4466  4473 
5104  5112 
5742  5751 

640 

641 

642 

643 

644 

645 

646 

647 

648  649 

1 
2 
3 

640 
1280 
1920 

641 
1282 
1923 

642 

1284 
1926 

643 
1286 
1929 

644 
1288 
1932 

645 
1290 
1935 

646 
1292 
1938 

647 
1294 
1941 

648  649 
1296  1298 
1944  1947 

4 
5 
6 

2560 
3200 
3840 

2564 
3205 
3846 

2568 
3210 
3852 

2572 
3215 
3858 

2576 
3220 
3864 

2580 
3225 
3870 

2584 
3230 
3876 

2588 
3235 
3882 

2592  2596 
3240  3245 
3888  3894 

7 

8 
9 

4480 
5120 
5760 

4487 
5128 
5769 

4494 
5136 

5778 

4501 
5144 

5787 

4508 
5152 
5796 

4515 
5160 
5805 

4522 
5168 
5814 

4529 
5176 

5823 

4536  4543 
5184  5192 
5832  5841 

XVIII.     MULTIPLES. 


61 


650 

651 

652 

653 

654 

655 

656 

657 

658  659 

1 

2 
3 

650 
1300 
1950 

651 
1302 
1953 

652 
1304 
1956 

653 
1306 
1959 

654 

1308 
1962 

655 
1310 
1965 

656 
1312 
1968 

657 
1314 
1971 

658  659 
1316  1318 
1974  1977 

4 
5 
6 

2600 
3250 
3900 

2604 
3255 
3906 

2608 
3260 
3912 

2612 
3265 
3918 

2616 
3270 
3924 

2620 
3275 
3930 

2624 
3280 
3936 

2628 
3285 
3942 

2632  2636 
3290  3295 
3948  3954 

7 
8 
9 

4550 
5200 
5850 

4557 
5208 
5859 

4564 
5216 
5868 

4571 
5224 
5877 

4578 
5232 
5886 

4585 
5240 
5895 

4592 
5248 
5904 

4599 
5256 
5913 

4606  4613 
5264  5272 
5922  5931 

660 

661 

662 

663 

664 

665 

666 

667 

668  669 

1 
2 
3 

660 
1320 
1980 

661 
1322 
1983 

662 
1324 
19S6 

663 
1326 
1989 

664 
1328 
1992 

665 
1330 

1995 

666 
1332 
1998 

667 
1334 
2001 

668  669 
1336  1338 
2004  2007 

4 
5 
6 

2640 
3300 
3960 

2644 
3305 
3966 

2648 
3310 
3972 

2652 
3315 
3978 

2656 
3320 
3984 

2660 
3325 
3990 

2664 
3330 
3996 

2668 
3335 
4002 

2672  2676 
3340  3345 
4008  4014 

7 
8 
9 

4620 
5280 
5940 

4627 
5288 
5949 

4634 
5296 
5958 

4641 
5304 
5967 

4648 
5312 
5976 

4655 
5320 

59S5 

4662 
5328 
5994 

4669 
5336 
6003 

4676  4683 
5344  5352 
6012  6021 

670 

671 

672 

673 

674 

675 

676 

677 

678  679 

1 
2 
3 

670 
1340 
2010 

671 
1342 
2013 

672 
1344 
2016 

673 
1346 
2019 

674 
1348 
2022 

675 
1350 
2025 

676 
1352 
2028 

677 
1354 
2031 

678  679 
1356  1358 
2034  2037 

4 
5 
6 

2680 
3350 
4020 

2684 
3355 
4026 

2688 
3360 
4032 

2692 
3365 
4038 

2696 
3370 
4044 

2700 
3375 
4050 

2704 
3380 
4056 

2708 
3385 
4062 

2712  2716 
3390  3395 
4068  4074 

7 
8 
9 

4690 
5360 
6030 

4697 
5368 
6039 

4704 
5376 
6048 

4711 

5384 
6057 

4718 
5392 
6066 

4725 
5400 
6075 

4732 
5408 
6084 

4739 
5416 
6093 

4746  4753 
5424  5432 
6102  6111 

680 

681 

682 

683 

684 

685 

686 

687 

688  689 

1 
3 

680 
1360 
2040 

681 
1362 
2043 

682 
1364 
2046 

683 
1366 
2049 

684 
1368 
2052 

685 
1370 
2055 

686 
1372 
2058 

687 
1374 
2061 

688  689 
1376  1378 
2064  2067 

4 
5 
6 

2720 
3400 
4080 

2724 
3405 
40S6 

2728 
3410 
4092 

2732 
3415 
4098 

2736 
3420 
4104 

2740 
3425 
4110 

2744 
3430 
4116 

2748 
3435 
4122 

2752  2756 
3440  3445 
4128  4134 

7 
8 
9 

4760 
5440 
6120 

4767 
5448 
6129 

4774 
5456 
6138 

4781 
5464 
6147 

4788 
5472 
6156 

4795 
5480 
6165 

4802 
5488 
6174 

4809 
5496 
6183 

4816  4823 
5504  5512 
6192  6201 

690 

691 

692 

693 

694 

695 

696 

697 

698  699 

1 
3 

690 
1380 
2070 

691 
1382 
2073 

692 
1384 
2076 

693 
1386 
2079 

694 
1388 
2082 

695 
1390 

2085 

696 
1392 

2088 

697 
1394 
2091 

698  699 
1396  1398 
2094  2097 

4 
5 
6 

2760 
3450 
4140 

2764 
3455 
4146 

2768 
3460 
4152 

2772 
3465 
4158 

2776 
3470 
4164 

2780 
3475 
4170 

2784 
3480 
4176 

2788 
3485 
4182 

2792  2796 
3490  3495 
4188  4194 

7 
8 
9 

4830 
5520 
6210 

4837 
5528 
6219 

4844 
5536 
6228 

4851 
5544 
6237 

4858 
5552 
6246 

4865 
5560 
6255 

4872 
5568 
6264 

4879 
5576 
6273 

4886  4893 
5584  5592 
6282  6291 

62 


XVIII.     MULTIPLES. 


1  700 

701 

702  703 

704 

706 

706 

707 

708  709 

1 

700 

701 

702   703 

704 

705 

706 

707 

708   709 

2 

1400 

1402 

1404  1406 

1408 

1410 

1412 

1414 

1416  1418 

3 

2100 

2103 

2106  2109 

2112 

2115 

2118 

2121 

2124  2127 

4 

2S00 

2804 

2808  2812 

2816 

2820 

2824 

2828 

2832  2836 

5 

3500 

3505 

3510  3515 

3520 

3525 

3530 

3535 

3540  3545 

6 

4200 

4206 

4212  4218 

4224 

4230 

4236 

4242 

4248  4254 

7 

4900 

4907 

4914  4921 

4928 

4935 

4942 

4949 

4956  4963 

8 

5600 

5608 

5616  5624 

5632 

5640 

5648 

5656 

5664  5672 

{> 

6300 

6309 

6318  6327 

6336 

6345 

6354 

6363 

6372  6381 

710 

711 

712  713 

714 

715 

71G 

717 

718  719 

1 

710 

711 

712  713 

714 

715 

716 

717 

718  719 

2 

1420 

1422 

1424  1426 

1428 

1430 

1432 

1434 

1436  1438 

3 

2130 

2133 

2136  2139 

2142 

2145 

2148 

2151 

2154  2157 

4 

2S40 

2844 

2848  2852 

2856 

2860 

2864 

2868 

2872  2876 

5 

3550 

3555 

3560  3565 

3570 

3575 

3580 

3585 

3590  3595 

G 

4260 

4266 

4272  4278 

4284 

4290 

4296 

4302 

4308  4314 

7 

4970 

4977 

4984  4991 

4998 

5005 

5012 

5019 

5026  5033 

H 

56S0 

5688 

5696  5704 

5712 

5720 

5728 

5736 

5744  5752 

\) 

6390 

6399 

6408  6417 

6426 

6435 

6444 

6453 

6462  6471 

720 

721 

722  723 

724 

725 

726 

727 

728  729 

1 

720 

721 

722   723 

724 

~^7 

726 

727 

728   729 

2 

1440 

1442 

1444  1446 

1448 

1450 

1452 

1454 

1456  1458 

3 

2160 

2163 

2166  2169 

2172 

2175 

2178 

2181 

2184  2187 

4 

28S0 

2884 

2888  2892 

2896 

2900 

2904 

2908 

2912  2916 

5 

3600 

3605 

3610  3615 

3620 

3625 

3630 

3635 

3640  3645 

6 

4320 

4326 

4332  4338 

4344 

4350 

4356 

4362 

4368  4374 

7 

5040 

5047 

5054  5061 

5068 

5075 

5082 

5089 

5096  5103 

H 

5760 

5768 

5776  5784 

5792 

5S00 

5808 

5816 

5824  5832 

9 

6480 

6489 

6498  6507 

6516 

6525 

6534 

6543 

6552  6561 

730 

731 

732  733 

734 

735 

736 

737 

738  739 

1 

730 

731 

732  733 

734 

735 

736 

737 

738   739 

2 

1460 

1462 

1464  1466 

1468 

1470 

1472 

1474 

1476  1478 

3 

2190 

2193 

2196  2199 

2202 

2205 

2208 

2211 

2214  2217 

4 

2920 

2924 

2928  2932 

2936 

2940 

2944 

2948 

2952  2956 

5 

3650 

3655 

3660  3665 

3670 

3675 

3680 

3685 

3690  3695 

G 

4380 

4386 

4392  4398 

4404 

4410 

4416 

4422 

4428  4434 

7 

5110 

5117 

5124  5131 

5138 

5145 

5152 

5159 

5166  5173 

8 

5840 

5848 

5856  5864 

5872 

5880 

5888 

5896 

5904  5912 

9 

6570 

6579 

6588  6597 

6606 

6615 

6624 

6633 

6642  6651 

740 

741 

742  743 

744 

745 

746 

747 

748  749 

1 

740 

741 

742   743 

744 

745 

746 

747 

748  749 

2 

1480 

1482 

1484  1486 

1488 

1490 

1492 

1494 

1496  1498 

3 

2220 

2223 

2226  2229 

2232 

2235 

2238 

2241 

2244  2247 

4 

2960 

2964 

2968  2972 

2976 

2980 

2984 

2988 

2992  2996 

5 

3700 

3705 

3710  3715 

3720 

3725 

3730 

3735 

3740  3745 

G 

4440 

4446 

4452  4458 

4464 

4470 

4476 

4482 

4488  4494 

7 

5180 

5187 

5194  5201 

5208 

5215 

5222 

5229 

5236  5243 

8 

5920 

5928 

5936  5944 

5952 

5960 

5968 

5976 

5984  5992 

9 

6660 

6669 

6678  6687 

6696 

6705 

6714 

6723 

6732  6741 

XVIII.     MULTIPLES. 


G3 


750 

751 

752  753 

754 

755 

756 

757 

758  759 

1 

2 
3 

750 
1500 
2250 

751 
1502 
2253 

752  753 
1504  1506 
2256  2259 

754 
1508 
2262 

755 
1510 

2265 

756 
1512 
2268 

757 
1514 
2271 

758  759 
1516  1518 
2274  2277 

4 
5 
6 

3000 
3750 
4500 

3004 
3755 
4506 

3008  3012 
3760  3765 
4512  4518 

3016 
3770 
4524 

3020 
3775 
4530 

3024 
3780 
4536 

3028 

3785 
4542 

3032  3036 
3790  3795 
4548  4554 

7 
8 
9 

5250 
6000 
6750 

5257 
6008 
6759 

5264  5271 
6016  6024 
6768  6777 

5278 
6032 
6786 

5285 
6040 
6795 

5292 
6048 
6804 

5299 
6056 
6813 

5306  5313 
6064  6072 
6822  6831 

760 

761 

762  763 

764 

765 

766 

767 

768  769 

1 
2 
3 

760 
1520 
2280 

761 
1522 
2283 

762   763 
1524  1526 
2286  2289 

764 
1528 
2292 

765 
1530 
2295 

766 
1532 
2298 

767 
1534 
2301 

768  769 
1536  1538 
2304  2307 

4 
5 

6 

3040 
3800 
4560 

3044 
3805 
4566 

3048  3052 
3810  3815 
4572  4578 

3056 
3820 
4584 

3060 
3825 
4590 

3064 
3830 
4596 

3068 
3835 
4602 

3072  3076 
3840  3845 
4608  4614 

7 

8 
9 

5320 
6080 
6840 

5327 
6088 
6849 

5334  5341 
6096  6104 
6858  6867 

5348 
6112 
6876 

5355 
6120 

6885 

5362 
6128 
6894 

5369 
6136 
6903 

5376  5383 
6144  6152 
6912  6921 

770 

771 

772  773 

774 

775 

776 

777 

778  779 

1 
2 
3 

770 
1540 
2310 

771 
1542 
2313 

772  773 
1544  1546 
2316  2319 

774 
1548 
2322 

775 
1550 
2325 

776 
1552 
2328 

777 
1554 
2331 

778  779 
1556  1558 
2334  2337 

4 
5 
6 

3080 
3850 
4620 

3084 

3855 
4626 

3088  3092 
3860  3865 
4632  4638 

3096 
3870 
4644 

3100 

3875 
4650 

3104 
3880 
4656 

3108 

3885 
4662 

3112  3116 
3890  3895 
4668  4674 

7 
8 
9 

5390 
6160 
6930 

5397 
6168 
6939 

5404  5411 
6176  6184 
6948  6957 

5418 
6192 
6966 

5425 
6200 
6975 

5432 
6208 
6984 

5439 
6216 
6993 

5446  5453 
6224  6232 
7002  7011 

780 

781 

782  783 

784 

785 

786 

787 

788  789 

1 
2 
3 

780 
1560 
2340 

781 
1562 
2343 

782  783 
1564  1566 
2346  2349 

784 
1568 
2352 

785 
1570 
2355 

786 
1572 
2358 

787 
1574 
2361 

788  789 
1576  1578 
2364  2367 

4 
5 
6 

3120 
3900 
4680 

3124 
3905 
4686 

3128  3132 
3910  3915 
4692  4698 

3136 
3920 
4704 

3140 
3925 
4710 

3144 
3930 
4716 

3148 
3935 
4722 

3152  3156 
3940  3945 
4728  4734 

7 

8 
9 

5460 
6240 
7020 

5467 
6248 
7029 

5474  5481 
6256  6264 
7038  7047 

5488 
6272 
7056 

5495 
6280 
7065 

5502 
6288 
7074 

5509 
6296 
7083 

5516  5523 
6304  6312 
7092  7101 

790 

791 

792  793 

794 

795 

796 

797 

798  799 

1 
2 

3 

790 
1580 
2370 

791 

1582 
2373 

792  793 
1584  1586 
2376  2379 

794 
1588 
2382 

795 
1590 

2385 

796 
1592 
2388 

797 
1594 
2391 

798  799 
1596  1598 
2394  2397 

4 
5 
6 

3160 
3950 
4740 

3164 
3955 
4746 

3168  3172 
3960  3965 
4752  4758 

3176 
3970 
4764 

3180 
3975 
4770 

3184 
3980 
4776 

3188 
3985 
4782 

3192  3196 
3990  3995 
4788  4794 

7 
8 
9 

5530 
6320 
7110 

5537 
6328 
7119 

5544  5551 
6336  6344 
7128  7137 

5558 
6352 
7146 

5565 
6360 

7155 

5572 
6368 
7164 

5579 
6376 
7173 

5586  5593 
6384  6392 
7182  7191 

64 


XVIII.     MULTIPLES. 


800 

801 

802 

803 

804 

805 

806 

807 

808 

809 

1 

800 

801 

802 

803 

804 

805 

806 

807 

808 

809 

2 

1600 

1602 

1604 

1606 

1608 

1610 

1612 

1614 

1616 

1618 

3 

2400 

2403 

2406 

2409 

2412 

2415 

2418 

2421 

2424 

2427 

4 

3200 

3204 

3208 

3212 

3216 

3220 

3224 

3228 

3232 

3236 

5 

4000 

4005 

4010 

4015 

4020 

4025 

4030 

4035 

4040 

4045 

6 

4S00 

4806 

4812 

4818 

4824 

4830 

4836 

4842 

4848 

4854 

7 

5600 

5607 

5614 

5621 

5628 

5635 

5642 

5649 

5656 

5663 

8 

6400 

6408 

6416 

6424 

6432 

6440 

6448 

6456 

6464 

6472 

9 

7200 

7209 

7218 

7227 

7236 

7245 

7254 

7263 

7272 

72S1 

810 

811 

812 

813 

814 

815 

816 

817 

818 

819 

1 

810 

811 

812 

813 

814 

815 

816 

817 

~81S" 

819 

2 

1620 

1622 

1624 

1626 

1628 

1630 

1632 

1634 

1636 

1638 

3 

2430 

2433 

2436 

2^39 

2442 

2445 

2448 

2451 

2454 

2457 

4 

3240 

3244 

3248 

3252 

3256 

3260 

3264 

3268 

3272 

3276 

5 

4050 

4055 

4060 

4065 

4070 

4075 

40S0 

4085 

4090 

4095 

6 

4S60 

4866 

4872 

4878 

4884 

4890 

4896 

4902 

4908 

4914 

7 

5670 

5677 

5684 

5691 

5698 

5705 

5712 

5719 

5726 

5733 

8 

6480 

6488 

6496 

6504 

6512 

6520 

6528 

6536 

6544 

6552 

9 

7290 

7299 

7308 

7317 

7326 

7335 

7344 

7353 

7362 

7371 

820 

821 

822 

823 

824 

825 

826 

827 

828 

829 

1 

820 

821 

822 

823 

824 

825 

826 

827 

828 

829 

2 

1640 

1642 

1644 

1646 

1648 

1650 

1652 

1654 

1656 

1658 

3 

2460 

2463 

2466 

2469 

2472 

2475 

2478 

2481 

2484 

2487 

4 

3280 

3284 

3288 

3292 

3296 

3300 

3304 

3308 

3312 

3316 

5 

4100 

4105 

4110 

4115 

4120 

4125 

4130 

4135 

4140 

4145 

6 

4920 

4926 

4932 

4938 

4944 

4950 

4956 

4962 

4968 

4974 

7 

5740 

5747 

5754 

5761 

5768 

5775 

5782 

5789 

5796 

5803 

8 

6560 

6568 

6576 

6584 

6592 

6600 

6608 

6616 

6624 

6632 

9 

73S0 

7389 

739S 

7407 

7416 

7425 

7434 

7443 

7452 

7461 

830 

831 

832 

833 

834 

835 

836 

837 

838 

839 

1 

830 

831 

832 

833 

834 

"835^ 

836 

837 

838 

839 

2 

1660 

1662 

1664 

1666 

1668 

1670 

1672 

1674 

1676 

1678 

3 

2490 

2493 

2496 

2499 

2502 

2505 

2508 

2511 

2514 

2517 

4 

3320 

3324 

3328 

3332 

3336 

3340 

3344 

3348 

3352 

3356 

5 

4150 

4155 

4160 

4165 

4170 

4175 

4180 

4185 

4190 

4195 

6 

4980 

4986 

4992 

4998 

5004 

5010 

5016 

5022 

5028 

5034 

7 

5810 

5817 

5824 

5831 

5838 

5845 

5852 

5859 

5866 

5873 

8 

6640 

6648 

6656 

6664 

6672 

6680 

6688 

6696 

6704 

4712 

9 

7470 

7479 

7488 

7^97 

7506 

7515 

7524 

7533 

7542 

7551 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

1 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

2 

1680 

1682 

1684 

1686 

1688 

1690 

1692 

1694 

1696 

1698 

3 

2520 

2523 

2526 

2529 

2532 

2535 

2538 

2541 

2544 

2547 

4 

3360 

3364 

3368 

3372 

3376 

3380 

3384 

3388 

3392 

3396 

5 

4200 

4205 

4210 

4215 

4220 

4225 

4230 

4235 

4240 

4245 

6 

5040 

5046 

5052 

5058 

5064 

5070 

5076 

5082 

5088 

5094 

7 

5880 

5887 

5894 

5901 

5908 

5915 

5922 

5929 

5936 

5943 

8 

6720 

6728 

6736 

6744 

6752 

6760 

6768 

6776 

6784 

6792 

9 

7560 

7569 

7578 

7587 

7596 

7605 

7614 

7623 

7632 

7641 

XVIII.     MULTIPLES. 


65 


850 

851 

852 

853 

854 

855 

856 

857 

858 

859 

1 

2 
3 

850 
1700 
2550 

851 
1702 

2553 

852 
1704 
2556 

853 
1706 
2559 

854 
1708 
2562 

855 
1710 
2565 

856 
1712 
2568 

857 
1714 
2571 

858 
1716 
2574 

859 
1718 

2577 

4 
5 

6 

3400 
4250 
5100 

3404 
4255 
5106 

3408 
4260 
5112 

3412 
4265 
5118 

3416 
4270 
5124 

3420 
4275 
5130 

3424 
4280 
5136 

3428 
4285 
5142 

3432 
4290 
5148 

3436 
4295 
5154 

7 

8 
9 

5950 
6800 
7650 

5957 
6808 
7659 

5964 
6816 
7668 

5971 
6824 
7677 

5978 
6832 
7686 

5985 
6840 
7695 

5992 
6848 
7704 

5999 
6856 
7713 

6006 
6864 

7722 

6013 
6872 
7731- 

860 

861 

862 

863 

864 

865 

866 

867 

868 

869 

1 

2 

3 

860 
1720 
2580 

861 
1722 

2583 

862 
1724 
2586 

863 
1726 
2589 

864 
1728 
2592 

865 
1730 
2595 

866 
1732 
2598 

867 
1734 
2601 

868 
1736 
2604 

869 
1738 
2607 

4 
5 
6 

3440 
4300 
5160 

3444 
4305 
5166 

3448 
4310 
5172 

3452 
4315 
5178 

3456 
4320 
5184 

3460 
4325 
5190 

3464 
4330 
5196 

3468 
4335 
5202 

3472 
4340 
5208 

3476 
4345 
5214 

7 
8 
9 

6020 
6880 
7740 

6027 
6888 
7749 

6034 
6896 

7758 

6041 
6904 
7767 

6048 
6912 
7776 

6055 
6920 

7785 

6062 
6928 
7794 

6069 
6936 
7803 

6076 
6944 

7812 

6083 
6952 
7821 

870 

871 

872 

873 

874 

875 

876 

877 

878 

879 

1 
2 
3 

870 
1740 
2610 

871 
1742 
2613 

872 
1744 
2616 

873 
1746 
2619 

874 
1748 
2622 

875 
1750 
2625 

876 
1752 
2628 

877 
1754 
2631 

878 
1756 
2634 

879 
1758 
2637 

4 
5 
6 

3480 
4350 
5220 

3484 
4355 
5226 

3488 
4360 
5232 

3492 
4365 
5238 

3496 
4370 
5244 

3500 
4375 
5250 

3504 
4380 
5256 

3508 
4385 
5262 

3512 
4390 
5268 

3516 
4395 
5274 

7 
8 
9 

6090 
6960 
7830 

6097 
6968 
7839 

6104 
6976 
7848 

6111 
6984 
7857 

6118 
6992 
7866 

6125 
7000 
7875 

6132 
7008 

7884 

6139 
7016 
7893 

6146 
7024 
7902 

6153 
7032 
7911 

880 

881 

882 

883 

884 

885 

886 

887 

888 

889 

1 
2 
3 

880 
1760 
2640 

881 
1762 
2643 

882 
1764 
2646 

883 
1766 
2649 

884 
1768 
2652 

885 
1770 
2655 

886 
1772 
2658 

887 
1774 
2661 

888 
1776 
2664 

889 
1778 
2667 

4 
5 
6 

3520 
4400 
5280 

3524 
4405 
5286 

3528 
4410 
5292 

3532 
4415 
5298 

3536 
4420 
5304 

3540 
4425 
5310 

3544 
4430 
5316 

3548 
4435 
5322 

3552 
4440 
5328 

3556 
4445 
5334 

7 
8 
9 

6160 
7040 
7920 

6167 

7048 
7929 

6174 
7056 
7938 

6181 
7064 
7947 

6188 
7072 
7956 

6195 
7080 
7965 

6202 
7088 
7974 

6209 
7096 
7983 

6216 
7104 
7992 

6223 
7112 
8001 

890 

891 

892 

893 

894 

895 

896 

897 

898 

899 

1 
2 
3 

890 
1780 
2670 

891 
1782 
2673 

892 
1784 
2676 

893 
1786 
2679 

894 
1788 
2682 

895 
1790 
2685 

896 
1792 
2688 

897 
1794 
2691 

898 
1796 
2694 

899 
1798 
2697 

4 
5 
6 

3560 
4450 
5340 

3564 
4455 
5346 

3568 
4460 

5352 

3572 
4465 
5358 

3576 
4470 
5364 

3580 
4475 
5370 

3584 
4480 
5376 

3588 
4485 

5382 

3592 
4490 
5388 

3596 
4495 
5394 

7 

8 
9 

6230 
7120 
8010 

6237 
7128 
8019 

6244 
7136 
8028 

6251 
7144 
8037 

6258 
7152 
8046 

6265 
7160 

8055 

6272 
7168 
8064 

6279 
7176 
8073 

6286 
7184 
8082 

6293 
7192 
8091 

Gfi 


XVIII.     MULTIPLES. 


900 

901 

902 

903 

904 

905 

906 

907 

908 

909 

1 

900 

901 

902 

903 

904 

905 

906 

907 

908 

909 

2 

ISOO 

1802 

1804 

1806 

1808 

1810 

1812 

1814 

1816 

1818 

3 

2700 

2703 

2706 

2709 

2712 

2715 

2718 

2721 

2724 

2727 

4 

3600 

3604 

3608 

3612 

3616 

3620 

3624 

3628 

3632 

3636 

5 

4500 

4505 

4510 

4515 

•1520 

4525 

4530 

4535 

4540 

4545 

6 

5400 

5406 

5412 

5418 

5424 

5430 

5436 

5442 

54-18 

5454 

7 

6300 

6307 

6314 

6321 

6328 

6335 

6342 

6349 

6356 

6363 

8 

7200 

7208 

7216 

7224 

7232 

7240 

7248 

7256 

7264 

7272 

9 

SlOO 

8109 

SllS 

8127 

8136 

8145 

8154 

8163 

8172 

8181 

910 

911 

912 

913 

914 

915 

916 

917 

918 

919 

1 

910 

911 

912 

913 

914 

915 

916 

917 

~918" 

919 

2 

1S20 

1822 

1824 

1826 

1828 

1830 

1832 

1834 

1836 

1838 

3 

2730 

2733 

2736 

2739 

2742 

2745 

2748 

2751 

2754 

2757 

4 

3640 

3644 

3648 

3652 

3656 

3660 

3664 

3668 

3672 

3676 

5 

4550 

4555 

4560 

4565 

4570 

4575 

4580 

4585 

4590 

4595 

6 

5460 

5466 

5472 

5478 

5484 

5490 

5496 

5502 

5508 

5514 

7 

6370 

6377 

6384 

6391 

6398 

6405 

6412 

6419 

6426 

6433 

8 

72S0 

7288 

7296 

7304 

7312 

7320 

7328 

7336 

7344 

7352 

9 

8190 

8199 

8208 

8217 

8226 

8235 

8244 

8253 

8262 

8271 

920 

921 

922 

923 

924 

925 

926 

927 

928 

929 

1 

920 

921 

922 

923 

924 

925 

926 

927 

928 

929 

2 

1840 

1842 

1844 

1846 

1848 

1850 

1852 

1854 

1856 

1858 

3 

2760 

2763 

2766 

2769 

2772 

2775 

2778 

2781 

2784 

2787 

4 

3680 

3684 

3688 

3692 

3696 

3700 

3704 

3708 

3712 

3716 

5 

4600 

4605 

4610 

4615 

4620 

4625 

4630 

4635 

4640 

4645 

6 

5520 

5526 

5532 

5538 

5544 

5550 

5556 

5562 

5568 

5574 

7 

6440 

6447 

6454 

6461 

6468 

6475 

6482 

6489 

6496 

6503 

8 

7360 

7368 

7376 

7384 

7392 

7400 

7-1  OS 

7416 

7424 

7432 

9 

8280 

8289 

8298 

8307 

8316 

8325 

8334 

8343 

8352 

8361 

930 

931 

932 

933 

934 

935 

936 

937 

938 

939 

1 

930 

931 

932 

933 

934 

935 

936 

937 

938 

939 

2 

1860 

1862 

1864 

1866 

1868 

1870 

1872 

1874 

1876 

1878 

3 

2790 

2793 

2796 

2799 

2802 

2805 

2808 

2811 

2814 

2817 

4 

3720 

3724 

3728 

3732 

3736 

3740 

37-14 

3748 

3752 

3756 

5 

4650 

4655 

4660 

4665 

4670 

4675 

4680 

4685 

4690 

4695 

6 

5580 

5586 

5592 

5598 

5604 

5610 

5616 

5622 

5628 

5634 

7 

6510 

6517 

6524 

6531 

6538 

6545 

6552 

6559 

6566 

6573 

8 

7440 

7448 

7456 

7464 

7472 

7480 

7488 

7496 

7504 

7512 

9 

8370 

8379 

8388 

8397 

8406 

8415 

8424 

8433 

8442 

8451 

940 

941 

942 

943 

944 

945 

946 

947 

948 

949 

1 

940 

941 

942 

943 

944 

945 

946 

947 

948 

949 

2 

1880 

1882 

1884 

1886 

ISSS 

1890 

1892 

1894 

1896 

1898 

3 

2820 

2823 

2826 

2829 

2832 

2835 

2838 

2841 

2844 

2847 

4 

3760 

3764 

3768 

3772 

3776 

3  780 

3784 

3788 

3792 

3796 

5 

4700 

4705 

4710 

4715 

4720 

4725 

4730 

4735 

4740 

4745 

6 

5640 

5646 

5652 

5658 

5664 

5670 

5676 

5682 

5688 

5694 

7 

6580 

6587 

6594 

6601 

6608 

6615 

6622 

6629 

6636 

6643 

8 

7520 

7528 

7536 

7544 

7552 

7560 

7568 

7576 

7584 

7592 

9 

8460 

8469 

8478 

8487 

84(1^ 

8505 

8514 

8523 

8532 

8541 

XVIII.    MULTIPLES. 


67 


950 

951 

952 

953 

954 

955 

956 

957 

958 

959 

1 

950 

951 

952 

953 

954 

955 

956 

957 

958 

959 

2 

1900 

1902 

1904 

1906 

1908 

1910 

1912 

1914 

1916 

1918 

3 

2850 

2853 

2856 

2859 

2862 

2865 

2868 

2871 

2874 

2877 

4 

3800 

3804 

3808 

3812 

3816 

3820 

3824 

3828 

3832 

3836 

5 

4750 

4755 

4760 

4765 

4770 

4775 

4780 

4785 

4790 

4795 

6 

5700 

5706 

5712 

571S 

5724 

5730 

5736 

5742 

5748 

5754 

7 

6650 

6657 

6664 

6671 

6678 

6685 

6692 

6699 

6706 

6713 

8 

7600 

7608 

7616 

7624 

7632 

7640 

7648 

7656 

7664 

7672 

9 

8550 

8559 

8568 

8577 

8586 

8595 

8604 

8613 

8622 

8631 

960 

961 

962 

963 

964 

965 

966 

967 

968 

969 

1 

960 

961 

962 

963 

964 

965 

966 

967 

968 

969 

2 

1920 

1922 

1924 

1926 

1928 

1930 

1932 

1934 

1936 

1938 

3 

2880 

2883 

2886 

2889 

2892 

2895 

2898 

2901 

2904 

2907 

4 

3840 

3844 

3848 

3852 

3856 

3860 

3864 

3868 

3872 

3876 

5 

4800 

4805 

4810 

4815 

4820 

4825 

4830 

483  5 

4840 

4845 

6 

5760 

5766 

5772 

5778 

5784 

5790 

5796 

5802 

5808 

5814 

7 

6720 

6727 

6734 

6741 

6748 

6755 

6762 

6769 

6776 

6783 

8 

7680 

7688 

7696 

7704 

7712 

7720 

7728 

7736 

7744 

7752 

9 

86+0 

8649 

8658 

8667 

8676 

8685 

8694 

8703 

8712 

8721 

970 

971 

972 

973 

974 

975 

976 

977 

978 

979 

1 

970 

971 

972 

973 

974 

975 

976 

977 

978 

979 

2 

1940 

1942 

1944 

1946 

1948 

1950 

1952 

1954 

1956 

1958 

3 

2910 

2913 

2916 

2919 

2922 

2925 

2928 

2931 

2934 

2937 

4 

3880 

3884 

3888 

3892 

3896 

3900 

3904 

3908 

3912 

3916 

5 

4850 

4855 

4860 

4865 

4870 

4875 

4SS0 

4885 

4890 

4895 

6 

5820 

5826 

5832 

5838 

5844 

5850 

5856 

5862 

5868 

5874 

7 

6790 

6797 

6804 

6811 

6818 

6825 

6832 

6839 

6846 

6853 

8 

7760 

7768 

7776 

7784 

7792 

7800 

7808 

7816 

7824 

7832 

9 

8730 

8739 

8748 

8757 

8766 

8775 

8784 

8793 

8802 

8811 

980 

981 

982 

983 

984 

985 

986 

987 

988 

989 

1 

980 

981 

982 

983 

984 

985 

986 

987 

~98S" 

989 

2 

1960 

1962 

1964 

1966 

1968 

1970 

1972 

1974 

1976 

1978 

3 

2940 

2943 

2946 

2949 

2952 

2955 

2958 

2961 

2964 

2967 

4 

3920 

3924 

3928 

3932 

3936 

3940 

3944 

3948 

3952 

3956 

5 

4900 

4905 

4910 

4915 

4920 

4925 

4930 

4935 

4940 

4945 

6 

5880 

5886 

5892 

5898 

5904 

5910 

5916 

5922 

5928 

5934 

7 

6860 

6867 

6874 

6881 

6888 

6895 

6902 

6909 

6916 

6923 

8 

7840 

7848 

7856 

7864 

7872 

7880 

7888 

7896 

7904 

7912 

9 

8820 

8829 

8S3S 

8847 

8856 

8865 

8874 

8883 

8892 

8901 

990 

991 

992 

993 

994 

995 

996 

997 

998 

999 

1 

990 

991 

992 

993 

994 

995 

996 

997 

998 

999 

2 

1980 

1982 

1984 

1986 

1988 

1990 

1992 

1994 

1996 

1998 

3 

2970 

2973 

2976 

2979 

2982 

2985 

2988 

2991 

2994 

2997 

4 

3960 

3964 

3968 

3972 

3976 

3980 

3984 

3988 

3992 

3996 

5 

4950 

4955 

4960 

4965 

4970 

4975 

4980 

4985 

4990 

4995 

6 

5940 

5946 

5952 

5958 

5964 

5970 

5976 

5982 

5988 

5994 

7 

6930 

6937 

6944 

6951 

6958 

6965 

6972 

6979 

6986 

6993 

8 

7920 

7928 

7936 

7944 

7952 

7960 

7968 

7976 

7984 

7992 

9 

8910 

8919 

8928 

8937 

8946 

8955 

8964 

8973 

8982 

8991 

68 


XIX.     CIROUMFERENOE   OF   OIROLE 


nn 


n 

0 

12        3 

4         5         6 

7         8        9 

PP 

1.0 

3.142 

3.173  3.204  3.236 

3.267  3.299  3.330 

3.362  3.393  3.424 

32 

1.1 
1.2 
1.3 

3.456 
3.770 
4.0S4 

3.4S7  3.519  3.550 
3.801  3.833  3.864 
4.115  4.147  4.178 

3.581  3.613  3.644 
3.896  3.927  3.958 
4.210  4.241  4.273 

3.676  3.707  3.738 
3.990  4.021  4.053 
4.304  4.335  4.367 

3 
6 
10 

1.4  4.39S 

1.5  4.712 

1.6  5.027 

4.430  4.461  4.492 
4.744  4.775  4.807 
5.058  5.089  5.121 

4.524  4.555  4.587 
4.838  4.869  4.901 
5.152  5.184  5.215 

4.618  4.650  4.681 
4.932  4.964  4.995 
5.246  5.278  5.309 

13 
16 
19 

1.7    5.341 

l.S    5.65i 
1.9  1  5.969 

5.372  5.404  5.435 
5.686  5.718  5.749 
6.000  6.032  6.063 

5.466  5.498  5.529 
5.781  5.812  5.843 
6.095  6.126  6.158 

5.561  5.592  5.623 
5.875  5.906  5.938 
6.189  6.220  6.252 

22 
26 
29 

2.0  i  6.2S3 

6.315  6.346  6.377 

6.409  6.440  6.472 

6.503  6.535  6.566 

31 

2.1 
2.2 
2.3 

6.597 
6.912 
7.226 

6.629  6.660  6.692 
6.943  6.974  7.006 
7.257  7.2SS  7.320 

6.723  6.754  6.786 
7.037  7.069  7.100 
7.351  7.383  7.414 

6.817  6.849  6.880 
7.131  7.163  7.194 
7.446  7.477  7.508 

3 
6 
9 

2.4 
2.5 
2.6 

7.540 
7.S54 
8.168 

7.571  7.603  7.634 
7.885  7.917  7.948 
8.200  8.231  8.262 

7.665  7.697  7.728 
7.980  8.011  8.042 
8.294  8.325  8.357 

7.760  7.791  7.823 
8.074  8.105  8.137 
8.388  8.419  8.451 

12 
16 
19 

2.7 

2.8 
2.9 

8.482 
8.796 
9.111 

8.514  8.545  8.577 
8.828  8.859  8.891 
9.142  9.173  9.205 

8.608  8.639  8.671 
8.922  8.954  8.985 
9.236  9.268  9.299 

8.702  8.734  8.765 
9.016  9.048  9.079 
9.331  9.362  9.393 

22 
25 
28 

3.0 

9.425 

9.456  9.488  9.519 

9.550  9.582  9.613 

9.645  9.676  9.708 

4 

3.1 
3.2 
3.3 

9.739 
10.05 
10.37 

9.770  9.802  9.833 
10.08  10.12  10.15 
10.40  10.43  10.46 

9.865  9.896  9.927 
10.18  10.21  10.24 
10.49  10.52  10.56 

9.959  9.990  10.02 
10.27  10.30  10.34 
10.59  10.62  10.65 

0 
1 
1 

3.4 
3.5 
3.6 

10.68 
11.00 
11.31 

10.71  10.74  10.78 
11.03  11.06  11.09 
11.34  11.37  11.40 

10.81  10.84  10.87 
11.12  11.15  11.18 
11.44  11.47  11.50 

10.90  10.93  10.96 
11.22  11.25  11.28 
11.53  11.56  11.59 

2 
2 
2 

3.7 
3.8 
3.9 

11.62 
11.94 
12.25 

11.66  11.69  11.72 
11.97  12.00  12.03 
12.28  12.32  12.35 

11.75  11.78  11.81 
12.06  12.10  12.13 
12.38  12.41  12.44 

11.84  11.88  11.91 
12.16  12.19  12.22 
12.47  12.50  12.53 

3 
3 

4 

4.0 

12.57 

12.60  12.63  12.66 

12.69  12.72  12.75 

12.79  12.82  12.85 

3 

4.1 
4.2 
4.3 

12.88 
13.19 
13.51 

12.91  12.94  12.97 
13.23  13.26  13.29 
13.54  13.57  13.60 

13.01  13.04  13.07 
13.32  13.35  13.38 
13.63  13.67  13.70 

13.10  13.13  13.16 
13.41  13.45  13.48 
13.73  13.76  13.79 

0 

1 
1 

4.4 
4.5 
4.6 

13.82 
14.14 
14.45 

13.85  13.89  13.92 
14.17  14.20  14.23 
14.48  14.51  14.55 

13.95  13.98  14.01 
14.26  14.29  14'33 
14.58  14.61  14.64 

14.04  14.07  14.11 
14.36  14.39  14.42 
14.67  14.70  14.73 

1 
2 
2 

4.7 

4.8 
4.9 

5.0 

14.77 
15.08 
15.39 

14.80  14.83  14.86 
15.11  15.14  15.17 
15.43  15.46  15.49 

14.89  14.92  14.95 
15.21  15.24  15.27 
15.52  15.55  15.58 

14.99  15.02  15.05 
15.30  15.33  15.36 
15.61  15.65  15.68 

2 
2 
3 

15.71 

15.74  15.77  15.80 

15.83  15.87  15.90 

15.93  15.96  15.99 

5.1 
5.2 
5.3 

16.02 
16.34 
16.65 

16.05  16.08  16.12 
16.37  16.40  16,43 
16.68  16.71  16.74 

16.15  16.18  16.21 
16.46  16.49  16.52 
16.78  16.81  16.84 

16.24  16.27  16.30 
16.56  16.59  16.62 
16.87  16.90  16.93 

5.4 
5.0 

5.6 

16.96 
17.28 
17.59 

17.00  17.03  17.06 
17.31  17.34  17.37 
17.62  17.66  17.69 

17.09  17.12  17.15 
17.40  17.44  17.47 
17.72  17.75  17.78 

17.18  17.22  17.25 
17.50  17.53  17.56 
17.81  17.84  17.88 

5.7 

5.8 
5.9 

17.91 
18.22 
18.54 

17.94  17.97  18.00 
18.25  18.28  18.32 
18.57  18.60  18.63 

18.03  18.06  18.10 
18.35  18.38  18.41 
18.66  18.69  18.72 

18.13  18.16  18.19 
18.44  18.47  18.50 
18.76  18.79  18.82 

im 


71  BEING   DIAMETER. 


69 


n 
6.0 

0 

12        3 

4        5        6 

7         8         9 

PP 

18.85 

18.88  18.91  18.94 

18.98  19.01  19.04 

19.07  19.10  19.13 

4 

6.1 

19.16 

19.20  19.23  19.26 

19.29  19.32  19.35 

19.38  19.42  19.45 

0 

6.2 

19.48 

19.51  19.54  19.57 

19.60  19.63  19.67 

19.70  19.73  19.76 

1 

6.3 

19.79 

19.82  19.85  19.89 

19.92  19.95  19.98 

20.01  20.04  20.07 

1 

6.4 

20.11 

20.14  20.17  20.20 

20.23  20.26  20.29 

20.33  20.36  20.39 

2 

6.5 

20.42 

20.45  20.48  20.51 

20.55  20.58  20.61 

20.64  20.67  20.70 

2 

6.6 

20.73 

20.77  20.80  20.83 

20.86  20.89  20.92 

20.95  20.99  21.02 

2 

6.7 

21.05 

21.08  21.11  21.14 

21.17  21.21  21.24 

21.27  21.30  21.33 

3 

6.8 

21.36 

21.39  21.43  21.46 

21.49  21.52  21.55 

21.58  21.61  21.65 

3 

6.9 

21.68 

21.71  21.74  21.77 

21.80  21.83  21.87 

21.90  21.93  21.96 

4 

7.0 

21.99 

22.02  22.05  22.09 

22.12  22.15  22.18 

22.21  22.24  22.27 

7.1 

22.31 

22.34  22.37  22.40 

22.43  22.46  22.49 

22.53  22.56  22.59 

7.2 

22.62 

22.65  22.68  22.71 

22.75  22.78  22.81 

22.84  22.87  22.90 

7.3 

22.93 

22.97  23.00  23.03 

23.06  23.09  23.12 

23.15  23.18  23.22 

7.4 

23.25 

23.28  23.31  23.34 

23.37  23.40  23.44 

23.47  23.50  23.53 

7.5 

23.56 

23.59  23.62  23.66 

23.69  23.72  23.75 

23.78  23.81  23.84 

7.6 

23.88 

23.91  23.94  23.97 

24.00  24.03  24.06 

24.10  24.13  24.16 

7.7 

24.19 

24.22  24.25  24.28 

24.32  24.35  24.38 

24.41  24.44  24.47 

7.8 

24.50 

24.54  24.57  24.60 

24.63  24.66  24.69 

24.72  24.76  24.79 

7.9 

24.82 

24.85  24.88  24.91 

24.94  24.98  25.01 

25.04  25.07  25.10 

8.0 

25.13 

25.16  25.20  25.23 

25.26  25.29  25.32 

25.35  25.38  25.42 

T 

8.1 

25.45 

25.48  25.51  25.54 

25.57  25.60  25.64 

25.67  25.70  25.73 

0 

8.2 

25.76 

25.79  25.82  25.86 

25.89  25.92  25.95 

25.98  26.01  26.04 

1 

8.3 

26.08 

26.11  26.14  26.17 

26.20  26.23  26.26 

26.30  26.33  26.36 

1 

8.4 

26.39 

26.42  26.45  26.48 

26.52  26.55  26.58 

26.61  26.64  26.67 

1 

8.5 

26.70 

26.73  26.77  26.80 

26.83  26.86  26.89 

26.92  26.95  26.99 

2 

8.6 

27.02 

27.05  27.08  27.11 

27.14  27.17  27.21 

27.24  27.27  27.30 

2 

8.7 

27.33 

27.36  27.39  27.43 

27.46  27.49  27.52 

27.55  27.58  27.61 

2 

8;8 

27.65 

27.68  27.71  27.74 

27.77  27.80  27.83 

27.87  27.90  27.93 

2 

8.9 

27.96 

27.99  28.02  28.05 

28.09  28.12  28.15 

28.18  28.21  28.24 

3 

9.0 

28.27 

28.31  28.34  28.37 

28.40  28.43  28.46 

28.49  28.53  28.56 

9.1 

28.59 

28.62  28.65  28.68 

28.71  28.75  28.78 

28.81  28.84  28.87 

9.2 

28.90 

28.93  28.97  29.00 

29.03  29.06  29.09 

29.12  29.15  29.19 

9.3 

29.22 

29.25  29.28  29.31 

29.34  29.37  29.41 

29.44  29.47  29.50 

9.4 

29.53 

29.56  29.59  29.63 

29.66  29.69  29.72 

29.75  29.78  29.81 

9.5 

29.85 

29.88  29.91  29.94 

29.97  30.00  30.03 

30.07  30.10  30.13 

9.6 

30.16 

30.19  30.22  30.25 

30.28  30.32  30.35 

30.38  30.41  30.44 

9.7 

30.47 

30.50  30.54  30.57 

30.60  30.63  30.66 

30.69  30.72  30.76 

9.8 

30.79 

30.82  30.85  30.88 

30,91  30.94  30.98 

31.01  31.04  31.07 

9.9 

31.10 

31.13  31.16  31.20 

31.23  31.26  31.29 

31.32  31.35  31.38 

value    log    recip. 

yfo 

0.0175  2.2419    57.2958 

TT 

3.1416  0.4971  0.3183 

■giro' 

0.0087  3.9408  114.5916 

27r 

6.2832  0.7982  0.1592 

TT- 

9.8696  0.9943      0.1013 

47r 

12.5664  1.0992  0.0796 

Vn 

1.7725  0.2486      0.5642 

7r 
2" 

1.5708  0.1961  0.6366 

ir^ 

31.006  1.4914      0.0323 

TT  =  3.1415926536 

f 

0.7854  1.8951  1.2732 

r. 

1.4646  0.1657      0.6828 

f 

0.5236  1.7190  1.9099 

Vi 

0.8060  1.9063      1.2407 

47r 
3 

4.1888  0.6221  0.2387 

W 

1.6120  0.2074      0.6204 

XX.     AEEA   or   OIKGLE 


n^ 


n 

0 

1       2       3 

4        5        G 

7         8        9 

d 

1.0 

0.785  1  0.801  0.817  0.833 

0.849  0.866  0.882 

0.899  0.916  0.933 

17 

1.1 
1.2 
1.3 

0.950 
1.131 
1.327 

0.968  0.985   1.003 
1.1.50  1.169  1.18S 
1.348  1.368  1.389 

1.021   1.039  1.057 
1.208  1.227  1.247 
1.410  1.431  1.453 

1.075  1.094  1.112 
1.267  1.287  1.307 
1.474  1.496  1.517 

19 
20 
22 

1.4 
1.5 
l.U 

1.539 
1.767 
2.011 

1.561  1..5S4  1.606 
1.791  1.815  1-839 
2.036  2.061  2.087 

1.629  1.651   1.674 
1.863  1.887  1.911 
2.112  2.138  2.164 

1.697  1.720  1.744 
1.936  1.961  1.986 
2.190-2.217  2.243 

23 

25 
27 

1.7 
1.8 
1.9 

2.270 
2.545 
2.835 

2.297  2.324  2.351 
2.573  2.602  2.6.30 
2.865  2.895  2.926 

2.378  2.405  2.433 
2.6.59  2.6SS  2.717 
2.956  2.986  3.017 

2.461  2.488  2.516 
2.746  2.776  2.806 
3.048  3.079  3.110 

29 
29 
32 

2.0  '3.142 

3.173  3.205  3.237 

3.269  3.301  3.333 

3.365  3.398  3.431 

33 

2.1  13.464 

2.2  '3.801 

2.3  !4.15i 

3.497  3.530  3.563 
3.8.36  3.871  3.906 
4.191  4.227  4.264 

3.597  3.631  3.664 
3.941  3.976  4.011 
4.301  4.337  4.374 

3.698  3.733  3.767 
4.047  4.083  4.119 
4.412  4.449  4.486 

34 
36 
38 

2.4 
2.5 
2.G 

4.524 
4.909 
5.309 

4.562  4.600  4.638 
4.948  4.9SS  5.027 
5.350  5.391  5.433 

4.676  4.714  4.753 
5.067  5.107  5.147 
5.474  5.515  5.557 

4.792  4.831  4.870 
5.187  5.228  5.269 
5.599  5.641  5.683 

39 
40 
43 

2.7 

2.8 
2.9 

5.726 
6.158 
6.605 

5.76S  5.811  5.S53 
6.202  6.246  6.290 
6.651  6.697  6.743 

5.896  5.940  5.983 
6.335  6.379  6.424 
6.789  6.835  6.881 

6.026  6.070  6.114 
6.469  6.514  6.560 
6.928  6.975  7.022 

44 
45 
47 

3.0 

7.069 

7.116  7.163  7.211 

7.258  7.306  7.354 

7.402  7.451  7.499 

49 

3.1 
3.2 
3.3 

7.548 
8.042 
8.553 

7.596  7.645  7.694 
8.093  8.143  8.194 
8.605  8.657  8.709 

7744  7.793  7.843 
8.245  8.296  8.347 
8.762  8.814  8.867 

7.892  7.942  7.992 
8.398  8.450  8..S01 
8.920  8.973  9.026 

50 
52 
53 

3.4 
3.5 
3.6 

9.079 
9.621 
10.18 

9.133  9.186  9.240 
9.676  9.731  9.787 
10.24  10.29  10.35 

9.294  9.348  9.402 
9.842  9.898  9.954 
10.41  10.46  10.52 

9.457  9.511  9.566 
10.01   10.07  10.12 
10.58  10.64  10.69 

55 
6 
6 

3.7  !  10.75 

3.8  11.34 

3.9  11.95 

10.81  10.87  10.93 
11.40  11.46  11.52 
12.01  12.07  12.13 

10.99  11.04  11.10 
11.58  11.64  11.70 
12.19  12.25  12.32 

11.16  11.22  11.28 
11.76  11.82  11.88 
12.38  12.44  12.50 

6 

7 
7 

4.0  1  12.57 

12.63  12.69  12.76 

12.82  12.88  12.95    13.01  13.07  13.14 

6 

4.1 
4.2 
4.3 

13.20 
13.85 
14.52 

13.27  13.33  13.40 
13.92  13.99  14.05 
14.59  14.66  14.73 

13.46  13.53  13.,59 
14.12  14.19  14.25 
14.79  14.86  14.93 

13.66  13.72  13.79 
14.32  14.39  14.45 
15.00  15.07  15.14 

6 

7 
7 

4.4 
4.5 
4.G 

15.21 
15.90 
16.62 

15.27  15.34  15.41 
15.98  16.05  16.12 
16.69  16.76  16.84 

15.48  15.55  15.62 
16.19  16.26  16.33 
16.91  16.98  17.06 

15.69  15.76  15.83 
16.40  16.47  16.55 
17.13  17.20  17.28 

7 
7 
7 

4.7  j  17.35 

4.8  '  18.10 

4.9  i  18.86 

17.42  17..S0  17.57 
18.17  18.25  18.32 
18.93  19.01  19.09 

17.65  17.72  17.80 
18.40  18.47  18.55 
19.17  19.24  19.32 

17.87  17.95   18.02 
18.63  18.70  18.78 
19.40  19.48  19.56 

8 
8 

7 

5.0  1  19.63 

19.71  19.79  19.87 

19.95  20.03  20.11 

20.19  20.27  20.35 

8 

5.1 
5.2 
5.3 

20.43 
21.24 
22.06 

20.51  20.59  20.67 
21.32  21.40  21.^8 
22.15  22.23  22.31 

20.75  20.83  20.91 
21.57  21.65  21.73 
22.40  22.48  22.56 

20.99  21.07  21.16 
21.81  21.90  21.98 
22.65  22.73  22.82 

8 
8 
8 

5.4 
5.5 
5.G 

22.90 
23.76 
24.63 

22.99  23.07  23.16 
23.84  23.93  24.02 
24.72  24.81  24.89 

23.24  23.33  23.41 
24.11  24.19  24.2S 
24.98  25.07  25.16 

23.50  23.59  23.67 
24.37  24.45  24.54 
25.25  25.34  25.43 

9 
9 
9 

5.7  =25.52 
5.S     26.42 
5.9     27.34 

25.61  25.70  25.79 
26.51   26.60  26.69 
27.43  27.53  27.62 

25.88  25.97  26.06 
26.79  26.  S8  26.97 
27.71  27.81  27.90 

26.15  26.24  26.33 
27.06  27.15  27.25 
27.99  28.09  28.18 

9 
9 
9 

n  BEING    DIAMETE3. 


71 


n 


3 


4       5 


8 


6.0 
6.1 
6.2 
6.3 
6.4 
6.5 
6.6 
6.7 
6.8 

6.y 

7.1 
7.2 
7.3 
7.4 
7.5 
7.6 
7.7 
7.8 
7.9 


8.0 
8.1 

8.2 
8.3 
8.4 
8.5 
8.6 
8.7 


28.27 
29.22 
30.19 
31.17 
32.17 
33.18 
34.21 
35.26 
36.32 
37.39 


28.37 
29.32 
30.29 
31.27 
32.27 
33.29 
34.32 
3.S.36 
36.42 
37.50 


28.46  28.56 
29.42  29.51 
30.39  30.48 
31.37  31.47 
32.37  32.47 
33.39  33.49 
34.42  34.52 

35.47  35.57 
36.53  36.64 
37.61  37.72 


28.65 
29.61 
30.58 
31.57 
32.57 
33.59 
34.63 
35.68 
36.75 
37.83 


28.75  28.84 
29.71  29.80 
30.68  30.78 
31.67  31.77 
32.67  32.78 
33.70  33.80 
34.73  34.84 
35.78  35.89 
36.85  36.96 
37.94  38.05 


28.94  29.03 
29.90  30.00 
30.88  30.97 

31.87  31.97 

32.88  32.98 
33.90  34.00 
34.94  35.05 
36.00  36.10 
37.07  37.18 
38.16  38.26 


29.13      9 
30.09  ,  10 


31.07 
32.07 


33.08  !  10 


34.11 
35.15 
36.21 

37.28 
38.37 


38.48 
39.59 
40.72 
41.85 
43.01 
44.18 
45.36 
46.57 
47.78 
49.02 


38.59  38.70 
39.70  39.82 
40.83  40.94 
41.97  42.08 
43.12  43.24 
44.30  44.41 
45.48  45.60 
46.69  46.81 
47.91  48.03 
49.14  49.27 


38.82 
39.93 
41.06 
42.20 
43.36 
44.53 
45.72 
46.93 
48.15 
49.39 


50.27 
51.53 
52.81 
54.11 
55.42 
56.75 
58.09 
59.45 
60.82 
62.21 


50.39 
51.66 
52.94 
54.24 

55.55 
56.88 
58.22 
59.58 
60.96 
62.35 


50.52  50.64 
51.78  51.91 
53.07  53.20 
54.37  54.50 
55.68  55.81 
57.01  57.15 
58.36  58.49 
59.72  59.86 
61.10  61.24 
62.49  62.63 


38.93 
40.04 
41.17 
42.31 
43.47 
44.65 
45.84 
47.05 
48.27 
49.51 


39.04  39.15 
40.15  40.26 
41.28  41.40 
42.43  42.54 
43.59  43.71 
44.77  44.89 
45.96  46.08 
47.17  47.29 
48.40  48.52 
49.64  49.76 


50.77 
52.04 
53.33 
54.63 
55.95 
57.28 
58.63 
59.99 
61.38 
62.77 


50.90  51.02 
52.17  52.30 
53.46  53.59 

54.76  54.89 
56.08  56.21 
57.41  57.55 

58.77  58.90 
60.13  60.27 
61.51  61.65 

62.91  63.05 


39.26 
40.38 
41.51 
42.66 
43.83 
45.01 
46.20 
47.42 
48.65 
49.89 


39.37 
40.49 
41.62 
42.78 
43.94 
45.13 
46.32 
47.54 
48.77 
50.01 


39.48    11 


40.60  ' 

41.74 

42.89 

44.06 

45.25 

46.45 

47.66 

48.89  ;  13 

50.14  ;  13 


51.15 
52.42 
53.72 
55.02 
56.35 
57.68 
59.04 
60.41 
61.79 
63.19 


51.28 
52.55 
53.85 
55.15 
56.48 
57.82 
59.17 
60.55 
61.93 
63.33 


51.40!  13 
52.68  13 
53.98  13 
55.29  i  13 
56.61  14 
57.95  I  14 
59.31  I  14 
60.68  14 
62.07  14 
63.48    14 


9.0 
9.1 
9.2 
9.3 
9.4 
9.5 
9.6 
9.7 
9.8 
9.9 


63.62 
65.04 
66.48 
67.93 
69.40 
70.88 
72.38 
73.90 
75.43 
76.98 


63.76 

65.18 
66.62 
68.08 
69.55 
71.03 
72.53 
74.05 
75.58 
77.13 


63.90  64.04 
65.33  65.47 
66.77  66.91 
68.22  68.37 
69.69  69.84 
71.18  71.33 
72.68  72.84 
74.20  74.36 
75.74  75.89 
77.29  77.44 


64.18 
65.61 
67.06 
68.51 
69.99 
71.48 
72.99 
74.51 
76.05 
77.60 


64.33  64.47 
65.76  65.90 
67.20  67.35 
68.66  68.81 
70.14  70.29 
71.63  71.78 
73.14  73.29 
74.66  74.82 
76.20  76.36 
77.76  77.91 


64.61 
66.04 
67.49 
68.96 
70.44 
71.93 
73.44 
74.97 
76.51 
7S.07 


64.75  64.90 
66.19  66.33 
67.64  67.78 
69.10  69.25 

70.58  70.73 
72.08  72.23 

73.59  73.75 
75.12  75.28 
76.67  76.82 
78.23  78.38 


32nds 

.03125 

.0625 

.09375 

.125 

.15625 

.1875 

.21875 

.25 

.28125 


36ths 

10 

.02(7) 

.0(5) 

.08(3) 

11 
12 
13 

•(1) 

.13(8) 

.1(6) 

14 
15 
16 

.19(4) 

.(2) 

.25 

17 

18 
19 

3125 

,34375 

,375 

,40625 

.4375 

.46875 

.5 

.53125 

.5625 

.59375 


.2(7) 

20 

.30(5) 

.(3) 

.36(1) 

21 
22 
23 

.3(8) 

24 
25 
26 

.47(2) 
.5 

27 

28 

.52(7) 

29 

.625 

.(5) 

30 

.65625 

..58(3) 

31 

.6875 

.6(1) 

32 

.71875 

.63(8) 

33 

.75 

.(6) 

34 

.78125 

.69(4) 

35 

.8125 

•7(2) 

36 

.84375 

.75 

.875 

.(7) 

.90625 

.80(5) 

.9375    .8(3) 
.96875  .86(1) 
.(8) 
.91(6) 
.9(4) 
.97(2) 


XXI.     CONTENT   OF   SPHEKE 


il3 


n 

0 

1         2         3     j     4         5         6 

7         8         9 

d 

1.0 

.5236 

.5395  .5556  .5722 

.5890  .6061  .6236 

.6414  .6596  .6781 

188 

1.1 
1.2 
l.*3 

.6969 
.9048 
1.1.50 

.7161  .7356  .7555 
.9276  .9508  .974.? 
1.177  1.204  1.232 

.7757  .7963  .8173 
.9983  1.023  1.047 
1.260  1.288  1.317 

.8386  .8603  .8823 
1.073  1.098  1.124 
1.346  1.376  1.406 

225 
26 
31 

1.4 '1.437 
1.5    1.767 
l.G    2.145 

1.468  1.499  1.531 
1.803  1.8.39  1.875 
2.185  2.226  2.268 

1.563  1.596  1.630 
1.912  1.9.50  1.988 
2.310  2.352  2.395 

1.663  1.697  1.732 
2.026  2.065  2.105 
2.439  2.483  2.527 

35 
40 
45 

1.7  2.572 

1.8  13.0.54 

1.9  3.591 

2.618  2.664  2.711 
3.105  3.157  3.209 
3.648  3.706  3.764 

2.758  2.806  2.855 
3.262  3.315  3.369 
3.823  3.882  3.942 

2.903  2.953  3.003 
3.424  3.479  3..535 
4.003  4.064  4.126 

51 
56 
63 

2.0   4.189 

4.252  4.316  4.380 

4.445  4.511  4.577 

4.644  4.712  4.780 

69 

2.1  4.849 

2.2  ,  5.575 

2.3  6.371 

4.919  4.989  5.060 
5.652  5.729  5.806 
6.454  6.538  6.623 

5.131  5.204  5.277 
5.885  5.964  6.044 
6.709  6.795  6.882 

5.350  5.425  5.500 
6.125  6.206  6.288 
6.970  7.059  7.148 

75 
83 
90 

2.4  7.238 

2.5  8.1  SI 
2.0  !  9.203 

7.329  7.421  7.513 
8.280  8.379  8.479 
9.309  9.417  9.525 

7.606  7.700  7.795 
8.580  8.682  8.785 
9.634  9.744  9.855 

7.890  7.986  8.083 
8.888  8.992  9.097 
9.966  10.08  10.19 

98 
106 
12 

2.7    10.31 

2..S'  11.49 
2.9    12.77 

10.42  10.54  10.65 
11.62  11.74  11.87 
12.90  13.04  13.17 

10.77  10.89  11.01 
11.99  12.12  12.25 
13.31   13.44  13.58 

11.13  11.25  11.37 
12.38  12.51  12.64 
13.72  13.86  14.00 

12 
13 
14 

3.0  1  14.14 

14.28  14.42  14.57 

14.71  14.86  15.00 

15.15  15.30  15.45 

15 

3.1  '  15.60 

3.2  17.16 

3.3  j  18.82 

15.75  15.90  16.06 
17.32  17.48  17.64 
18.99  19.16  19.33 

16.21  16.37  16.52 
17.81  17.97  18.14 
19.51  19.68  19.86 

16.68  16.84  17.00 
18.31  18.48  18.65 
20.04  20.22  20.40 

16 
17 
18 

3.4  20.58 

3.5  1  22.45 

3.6  1  24.43 

20.76  20.94  21.13 
22.64  22.84  23.03 
24.63  24.84  25.04 

21.31  21.50  21.69 
23.23  23.43  23.62 
25.25  25.46  25.67 

21.88  22.07  22.26 
23.82  24.02  24.23 
25.88  26.09  26.31 

19 

20 
21 

3.7 
3.8 

3.9 

26.52 
28.73 
31.06 

26.74  26.95  27.17 
28.96  29.19  29.42 
31.30  31.54  31.78 

27.39  27.61  27.83 
29.65  29.88  30.11 
32.02  32.27  32.52 

28.06  28.28  28.50 
30.35  30.58  30.82 
32.76  33.01  33.26 

23 
24 
25 

4.0 

33.51 

33.76  34.02  34.27 

34.53  34.78  35.04 

35.30  35.56  35.82 

27 

4.1 
4.2 
4.3 

36.09 
38.79 
41.63 

36.35  36.62  36.88 
39.07  39.35  39.63 
41.92  42.21  42.51 

37.15  37.42  37.69 
39.91  40.19  40.48 
42.80  43.10  43.40 

37.97  38.24  38.52 
40.76  41.05  41.34 
43.70  44.00  44.30 

27 
29 
30 

4.4 
4.5 
4.G 

44.60 
47.71 
50.97 

44.91  45.21  45.52 
48.03  48.35  48.67 
51.30  51.63  51.97 

45.83  46.14  46.45 
49.00  49.32  49.65 
52.31  52.65  52.99 

46.77  47.08  47.40 
49.97  50.30  50.63 
53.33  53.67  54.02 

31 
34 
34 

4.7 

4.8 
4.9 

54.36 
57.91 
61.60 

54.71  55.06  55.41 
58.27  58.63  59.00 
61.98  62.36  62.74 

55.76  56.12  56.47 
59.37  59.73  60.10 
63.12  63.51  63.89 

56.83  57.19  57.54 
60.48  60.85  61.22 
64.28  64.67  65.06 

37 
38 
39 

5.0 

65.45 

65.84  66.24  66.64 

67.03  67.43  67.83 

68.24  68.64  69.05 

41 

5.1 
5.2 
5.3 

69.46 

73.62 
77.95 

69.87  70.28  70.69 
74.05  74.47  74.90 
78.39  78.84  79.28 

71.10  71.52  71.94 
75.33  75.77  76.20 
79.73  80.18  80.63 

72.36  72.78  73.20 
76.64  77.07  77.51 
81.08  81.54  81.99 

42 
44 
46 

5.4 
5.5 
5.6 

82.45 
87.11 
91.95 

82.91  83.37  83.83 
87.59  88.07  88.55 
92.45  92.94  93.44 

84.29  84.76  85.23 
89.03  89.51  90.00 
93.94  94.44  94.94 

85.70  86.17  86.64 
90.48  90.97  91.46 
95.44  95.95  96.46 

47 
49 
51 

5.7  '  96.97 

5.8  102.2 

5.9  107.5 

97.48  97.99  98.51 
102.7  103.2  103.8 
108.1  108.6  109.2 

99.02  99.54  100.1 
104.3  104.8  105.4 
109.7  110.3  110.9 

100.6  101.1  101.6 
105.9  106.4  107.0 
111.4  112.0  112.5 

6 

5 
6 

^^3 


n  BEING    DIAMETER. 


73 


6.0 
.1 
6.2 
6.3 
J.4 
6.5 
6.6 
6.7 
6.8 
6.9 


113.1 
11S.8 
124.8 
130.9 
137.3 
143.8 
150.5 
157.5 
164.6 
172.0 


113.7 
119.4 
125.4 
131.5 
137.9 
144.5 
151.2 
158.2 
165.4 
172.8 


114.2  114.8 
120.0  120.6 

126.0  126.6 
132.2  132.8 
138.5  139.2 

145.1  145.8 
151.9  152.6 
158.9  159.6 
166.1  166.8 
173.5  174.3 


115.4 
121.2 

127.2 
133.4 
139.8 
146.5 
153.3 
160.3 
167.6 
175.0 


115.9  116.5 
121.8  122.4 
127.8  128.4 
134.1  134.7 
140.5  141.2 
147.1  147.8 
154.0  154.7 
161.0  161.7 
168.3  169.0 
175.8  176.5 


117.1 
123.0 
129.1 
135.3 
141.8 
14S.5 
155.4 
162.5 
169.8 
177.3 


117.7  118.3 

123.6  124.2 

129.7  130.3 

136.0  136.6 
142.5  143.1 
149.2  149.8 

156.1  156.8 

163.2  163.9 
170.5  171.3 
178.1  178.8 


7.0 
7.1 
7.2 
7.3 
7.4 
7.5 
7.6 
7.7 
7.8 
7.9 


179.6 
187.4 
195.4 
203.7 
212.2 
220.9 
229.8 
239.0 
248.5 
258.2 


180.4 
1SS.2 
196.2 
204.5 
213.0 
221.8 
230.8 
240.0 
249.4 
259.1 


181.1  181.9 

189.0  189.8 

197.1  197.9 
205.4  206.2 
213.9  214.8 
222.7  223.6 
231.7  232.6 
240.9  241.8 
250.4  251.4 
260.1  261.1 


182.7 
190.6 
198.7 
207.1 
215.6 
224.4 
233.5 
242.8 
252.3 
262.1 


183.5  184.3 

191.4  192.2 

199.5  200.4 
207.9  208.8 
216.5  217.4 

225.3  226.2 

234.4  235.3 
243.7  244.7 
253.3  254.3 
263.1  264.1 


185.0 
193.0 
201.2 
209.6 
218.3 
227.1 
236.3 
245.6 
255.2 
265.1 


185.8  186.6 
193.8  194.6 

202.0  202.9 

210.5  211.3 

219.1  220.0 

228.0  228.9 

237.2  238.1 

246.6  247.5 
256.2  257.2 

266.1  267.1 


8.0 

8.1 

8.2 

.3 

A 


8.7 
8.8 


268.1 
278.3 
288.7 
299.4 
310.3 
321.6 
333.0 
344.8 
356.8 
369.1 


269.1 
279.3 
289.8 
300.5 
311.4 
322.7 
334.2 
346.0 
358.0 
370.4 


270.1  271.1 

280.3  281.4 
290.8  291.9 
301.6  302.6 
312.6  313.7 
323.8  325.0 

335.4  336.5 

347.2  348.4 

359.3  360.5 
371.6  372.9 


272.1 
282.4 
292.9 
303.7 
314.8 
326.1 
337.7 
349.6 
361.7 
374.1 


273.1  274.2 
283.4  284.5 
294.0  295.1 

304.8  305.9 

315.9  317.0 

327.3  328.4 
338.9  340.1 
3.50.8  352.0 
362.9  364.2 

375.4  376.6 


275.2 
285.5 
296.2 
307.0 
318.2 
329.6 
341.2 
353.2 
365.4 
377.9 


276.2 
286.6 
297.2 
308.1 
319.3 
330.7 
342.4 
354.4 
366.6 
379.2 


277.2 
287.6 
298.3 
309.2 
320.4 
331.9 
343.6 
355.6 
367.9 
380.4 


9.0 
9.1 
9.2 
9.3 
9.4 
9.5 
9.6 
9.7 
9.8 
9.9 


381.7 
394.6 
407.7 
421.2 
434.9 
448.9 
463.2 
477.9 
492.8 
508.0 


383.0 
395.9 
409.1 
422.5 
436.3 
450.3 
464.7 
479.4 
494.3 
509.6 


384.3  385.5 
397.2  398.5 

410.4  411.7 
423.9  425.2 

437.7  439.1 

451.8  453.2 
466.1  467.6 
480.8  482.3 
495.8  497.3 
511.1  512.7 


386.8  388.1 
399.8  401.1 
413.1  414.4 
426.6  428.0 

440.5  441.9 

454.6  456.0 

469.1  470.5 
483.8  485.3 
•198.9  500.4 

514.2  515.8 


389.4 
402.4 
415.7 
429.4 
443.3 
457.5 
472.0 
486.8 
501.9 
517.3 


390.7 
403.7 
417.1 
430.7 
444.7 
458.9 
473.5 
488.3 
503.4 
518.9 


392.0 
405.1 
418.4 
432.1 
446.1 
460.4 
474.9 
489.8 
505.0 
520.5 


393.3 
406.4 
419.8 
433.5 
447.5 
461.8 
476.4 
491.3 
506.5 
522.0 


log  n  ! 

0.0000 
0.3010 
0.7782 
1.3802 
2.0792 
2.8573 
3.7024 
4.6055 
5.5598 


log  2» 

0.3010 
0.6021 
0.9031 
1.2041 
1.5051 
1.8062 
2.1072 
2.4082 
2.7093 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


6.5598 
7.6012 
8.6803 
9.7943 
10.9404 
12.1165 
13.3206 
14.5511 
15.8063 
17.0851 


3.0103 

20 

3.3113 
3.6124 
3.9134 

21 
22 
23 

4.2144 
4.5154 
4.8165 

24 
25 
26 

5.1175 

5.4185 
5.7196 

27 

28 
29 

18.3861 
19.7083 
21.0508 
22.4125 
23.7927 
25.1906 
26.6056 
28.0370 
29.4841 
30.9465 


6.0206 
6.3216 
6.6227 
6.9237 
7.2247 
7.5257 
7.8268 
8.1278 
8.4288 
8.7299 


74 


XXII.     HYPERBOLIC   LOGAEITHMS. 


logen 


n\     0 

1    2   3 

4   5   6 

7    8    9 

d 

i.o'o.oooo 

0100  0198  0296 

0392  0488  0583 

0677  0770  0862 

91 

1.1  1  0953 

1.2  1S23 

1.3  2624 

1044  1133  1222 
1906  1989  2070 
2700  2776  2852 

1310  1398  1484 
2151  2231  2311 
2927  3001  3075 

1570  1655  1740 
2390  2469  2546 
3148  3221  3293 

83 
78 
72 

1.4  336i 
1.5 i  4055 
1.61  4700 

3436  3507  3577 
4121  4187  4253 
4762  4824  4886 

3646  3716  3784 
4318  4383  4447 
4947  5008  5068 

3853  3920  3988 
4511  4574  4637 
5128  5188  5247 

67 
63 
59 

1.7 

1.8 
1.9 

5306 
5S78 
6419 

5365  5423  5481 
5933  5988  6043 
6471  6523  6575 

5539  5596  5653 
6098  6152  6206 
6627  6678  6729 

5710  5766  5822 
6259  6313  6366 
6780  6831  6881 

56 
53 
50 

2.0 

0.6931 

6981  7031  7080 

7129  7178  7227 

7275  7324  7372 

47 

2.1 
2.2 
2.3 

7419 

7SS5 
8329 

7467  7514  7561 
7930  7975  8020 
8372  8416  8459 

7608  7655  7701 
8065  8109  8154 
8502  8544  8587 

7747  7793  7839 
8198  8242  8286 
8629  8671  8713 

46 
43 
42 

2.4 
2.5 
2.6 

8755 
9163 
9555 

8796  8838  8879 
9203  9243  9282 
9594  9632  9670 

8920  8961  9002 
9322  9361  9400 
9708  9746  9783 

9042  9083  9123 
9439  9478  9517 
9821  9858  9895 

40 
38 
38 

2.7 

2.8 
2.9 

9933 

1.0296 

0647 

9969*0006*0043 
0332  0367  0403 
0682  0716  0750 

*0080*01 16*0152 
0438  0473  0508 
0784  0818  0852 

*0188*0225*0260 
0543  0578  0613 
0886  0919  0953 

36 
34 
33 

3.0 

1.09S6 

1019  1053  1086 

1119  1151  1184 

1217  1249  1282 

32 

3.1 
3.2 
3.3 

1314 
1632 
1939 

1346  1378  1410 
1663  1694  1725 
1969  2000  2030 

1442  1474  1506 
1756  1787  1817 
2060  2090  2119 

1537  1569  1600 
1848  1878  1909 
2149  2179  2208 

32 
30 
30 

3.4 
3.5 
3.6 

2238 
2528 
2S09 

2267  2296  2326 
2556  2585  2613 
2837  2865  2892 

2355  2384  2413 
2641  2669  2698 
2920  2947  2975 

2442  2470  2499 
2726  2754  2782 
3002  3029  3056 

29 

27 
27 

3.7 
3.8 
3.9 

3083 
3350 
3610 

3110  3137  3164 
3376  3403  3429 
3635  3661  3686 

3191  3218  3244 
3455  3481  3507 
3712  3737  3762 

3271  3297  3324 
3533  3558  3584 
3788  3813  3838 

26 
26 

25 

4.0 

1.3863 

3888  3913  3938 

3962  3987  4012 

4036  4061  4085 

25 

4.1 
4.2 
4.3 

4110 
4351 
4586 

4134  4159  4183 
4375  4398  4422 
4609  4633  4656 

4207  4231  4255 
4446  4469  4493 
4679  4702  4725 

4279  4303  4327 
4516  4540  4563 
4748  4770  4793 

24 
23 
23 

4.4 
4.5 
4.6 

4S16 
5041 
5261 

4839  4861  4884 
5063  5085  5107 
5282  5304  5326 

4907  4929  4951 
5129  5151  5173 
5347  5369  5390 

4974  4996  5019 
5195  5217  5239 
5412  5433  5454 

22 
22 
22 

4.7 

4.8 
4.9 

5476 
5686 
5892 

5497  5518  5539 
5707  5728  5748 
5913  5933  5953 

5560  5581  5602 
5769  5790  5810 
5974  5994  6014 

5623  5644  5665 
5831  5851  5872 
6034  6054  6074 

21 

20 
20 

5.0 

1.6094 

6114  6134  6154 

6174  6194  6214 

6233  6253  6273 

19 

5.1 
5.2 
5.3 

6292 
6487 
6677 

6312  6332  6351 
6506  6525  6544 
6696  6715  6734 

6371  6390  6409 
6563  6582  6601 
6752  6771  6790 

6429  6448  6467 
6620  6639  6658 
6808  6827  6845 

20 
19 
19 

5.4 
5.5 
5.6 

6864 
7047 
7228 

6882  6901  6919 
7066  7084  7102 
7246  7263  7281 

6938  6956  6974 
7120  7138  71.56 
7299  7317  7334 

6993  7011  7029 
7174  7192  7210 
7352  7370  7387 

18 
18 
18 

5.7 

5.8 
5.9 

7405 
7579 
7750 

7422  7440  7457 
7596  7613  7630 
7766  7783  7800 

7475  7492  7509 
7647  7664  7681 
7817  7834  7851 

7527  7544  7561 
7699  7716  7733 
7867  7884  7901 

18 
17 
17 

lOge^i 


XXII.    HYPEEBOLIO   LOGAEITHMS. 


n 

0 

12   3 

4   5   6 

7    8    9 

d 

6.0 

1.7918 

7934  7951  7967 

7984  8001  8017 

8034  8050  8066 

17 

6.1 
6.2 
6.3 

8083 
8245 
8405 

8099  8116  8132 
8262  8278  8294 
8421  8437  8453 

8148  8165  8181 
8310  8326  8342 
8469  8485  8500 

8197  8213  8229 
8358  8374  8390 
8516  8532  8547 

16 
15 
16 

6.4 
6.5 
6.6 

8563 
8718 

8871 

8579  8594  8610 
8733  8749  8764 
8886  8901  8916 

8625  8641  8656 
8779  8795  8810 
8931  8946  8961 

8672  8687  8703 
8825  8840  8856 
8976  8991  9006 

15 
15 
15 

6.7 
6.8 
6.9 

7.0 

9021 
9169 
9315 

9036  9051  9066 
9184  9199  9213 
9330  9344  9359 

9081  9095  9110 
9228  9242  9257 
9373  9387  9402 

9125  9140  9155 
9272  9286  9301 
9416  9430  9445 

14 
14 
14 

1.9459 

9473  9488  9502 

9516  9530  9544 

9559  9573  9587 

14 

7.1 
7.2 
7.3 

9601 
9741 
9879 

9615  9629  9643 
9755  9769  9782 
9892  9906  9920 

9657  9671  9685 
9796  9810  9824 
9933  9947  9961 

9699  9713  9727 
9838  9851  9865 
9974  9988*0001 

14 
14 
14 

7.4 
7.5 

7.6 

2.0015 
0149 
0281 

0028  0042  0055 
0162  0176  0189 
0295  0308  0321 

0069  0082  0096 
0202  0215  0229 
0334  0347  0360 

0109  0122  0136 
0242  0255  0268 
0373  0386  0399 

13 
13 
13 

7.7 
7.8 
7.9 

0412 
0541 
0669 

0425  0438  0451 
0554  0567  0580 
0681  0694  0707 

0464  0477  0490 
0592  0605  0618 
0719  0732  0744 

0503  0516  0528 
0631  0643  0656 
0757  0769  0782 

13 

13 
12 

8.0 

2.0794 

0807  0819  0832 

0844  0857  0869 

0882  0894  0906 

13 

8.1 

8.2 
8.3 

0919 
1041 
1163 

0931  0943  0956 
1054  1066  1078 
1175  1187  1199 

0968  0980  0992 
1090  1102  1114 
1211  1223  1235 

1005  1017  1029 
1126  1138  1150 
1247  1258  1270 

12 
13 
12 

8.4 
8.5 
8.6 

1282 
1401 
1518 

1294  1306  1318 
1412  1424  1436 
1529  1541  1552 

1330  1342  ]353 
1448  1459  1471 
1564  1576  1587 

1365  1377  1389 
1483  1494  1506 
1599  1610  1622 

12 
12 
11 

8.7 
8.8 
8.9 

1633 
1748 
1861 

1645  1656  1668 
1759  1770  1782 
1872  1883  1894 

1679  1691  1702 
1793  1804  1815 
1905  1917  1928 

1713  1725  1736 
1827  1838  1849 
1939  1950  1961 

12 
12 
11 

9.0 

2.1972 

1983  1994  2006 

2017  2028  2039 

2050  2061  2072 

11 

9.1 
9.2 
9.3 

2083 
2192 
2300 

2094  2105  2116 
2203  2214  2225 
2311  2322  2332 

2127  2138  2148 
2235  2246  2257 
2343  2354  2364 

2159  2170  2181 
2268  2279  2289 
2375  2386  2396 

11 
11 
11 

9.4 
9.5 
9.6 

2407 
2513 
2618 

2418  2428  2439 
2523  2534  2544 
2628  2638  2649 

2450  2460  2471 
2555  2565  2576 
2659  2670  2680 

2481  2492  2502 
2586  2597  2607 
2690  2701  2711 

11 
11 

10 

9.7 
9.8 
9.9 

2721 
2824 
2925 

2732  2742  2752 
2834  2844  2854 
2935  2946  2956 

2762  2773  2783 
2865  2875  2885 
2966  2976  2986 

2793  2803  2814 
2895  2905  2915 
2996  3006  3016 

10 
10 
10 

n 

1 
in 

1st   2d    S 

A       4th 

in 

logelO'V  logelO-« 

1 

2.3026 

0.2303  0.0230  O.C 

023  0.0002 

0.4343 

2.3026   3.6974 

2 

4.6052 

0.4605  0.0461  0.0 

046  0.0005 

0.8686 

4.6052   5.3948 

3 

6.9078 

0.6908  0.0691  0.0 

069  0.0007 

1.3029 

6.9078   7.0922 

4 

9.2103 

0.9210  0.0921  0.0 

092  0.0009 

1.7372 

9.2103   10.7897 

5 

11.5129 

1.1513  0.1151  O.C 

115  0.0012 

2.1715 

11.5129   12.4871 

6 

13.8155 

1.3816  0.1382  O.C 

138  0.0014 

2.6058 

13.8155   14.1845 

7 

16.1181 

1.6118  0.1612  O.C 

161  0.0016 

3.0401 

16.1181   17.8819 

8 

18.4207 

1.8421  0.1842  O.C 

184  0.0018 

3.4744 

18.4207   19.5793 

9 

20.7233 

2.0723  0.2072  O.C 

207  0.0021 

3.9087 

20.7233   21.2767 

JC,    XXIII.   AMOUNT  AT  THE  END  OF  u  YEAES. 


.        WO 


n 

3       3A       4 

4}       6       6 

7         8        9 

10 

0 

1.000  1.000  1.000 

1.000  1.000  1.000 

1.000  1.000  1.000 

1.000 

1 

1.030  1.03.^  1.040 

1.045  1.050  1.060 

1.070  1.080  1.090 

1.100 

o 

1.061   1.071   1.0S2 

1.092  1.103  1.124 

1.145  1.166  1.188 

1.210 

8 

1.093  1.109  1.125 

1.141  1.158  1.191 

1.225  1.260  1.295 

1.331 

4 

1.126  1.H8  1.170 

1.193  1.216  1.262 

1.311  1.360  1.412 

1.464 

5 

1.1-S9  1.1S8  1.217 

1.246  1.276  1.338 

1.403  1.469  1.539 

1.611 

6 

1.194  1.229  1.265 

1.302  1.340  1.419 

1.501  1.587  1.677 

1.772 

7 

1.230  1.272  1.316 

1.361   1.407  1..504 

1.606  1.714  1.828 

1.949 

8 

1.267  1.317  1.369 

1.422  1.477  1.594 

1.718  1.851  1.993 

2.144 

.     9 

1.30i  1.363  1.423 

1.486  1.551  1.689 

1.838  1.999  2.172 

2.358 

10 

1.344  1.411  1.480 

1.5.53  1.629  1.791 

1.967  2.159  2.367 

2.594 

11 

1.384  1.460  1..S39 

1.623  1.710  1.S9S 

2.105  2.332  2.580 

2.853 

12 

1.426  1..S11   1.601 

1.696  1.796  2.012 

2.252  2.518  2.813 

3.138 

13 

1.469  1.564  1.665 

1.772  1.886  2.133 

2.410  2.720  3.066 

3.452 

U 

1.513  1.619  1.732 

1.852  1.980  2.261 

2.579  2.937  3.342 

3.797 

15 

1.558  1.675  1.801 

1.935  2.079  2.397 

2.759  3.172  3.642 

4.177 

IG 

1.605  1.734  1.873 

2.022  2.183  2..540 

2.952  3.426  3.970 

4.595 

17 

1.653  1.795  1.948 

2.113  2.292  2.693 

3.159  3.700  4.328 

5.054 

18 

1.702  1.857  2.026 

2.208  2.407  2.854 

3.380  3.996  4.717 

5.560 

19 

1.754  1.923  2.107 

2.308  2.527  3.026 

3.617  4.316  5.142 

6.116 

20" 

1.806  1.990  2.191 

2.412  2.653  3.207 

3.870  4.661  5.604 

6.727 

21 

1.860  2.059  2.279 

2.520  2.786  3.400 

4.141  5.034  6.109 

7.400 

.).> 

1.916  2.132  2.370 

2.634  2.925  3.604 

4.430  5.437  6.659 

8.140 

23 

1.974  2.206  2.465 

2.752  3.072  3.820 

4.741  5.871  7.258 

8.954 

24 

2.033  2.283  2.563 

2.876  3.225  4.049 

5.072  6.341  7.911 

9.850 

25 

2.094  2.363  2.666 

3.005  3.386  4.292 

5.427  6.848  8.623 

10.83 

20 

2.157  2.446  2.772 

3.141  3.556  4.549 

5.807  7.396  9.399 

11.92 

27 

2.221  2.532  2.883 

3.282  3.733  4.822 

6.214  7.988  10.25  . 

13.11 

28 

2.288  2.620  2.999 

3.430  3.920  5.112 

6.649  8.627  11.17 

14.42 

29 

2.357  2.712  3.119 

3.584  4.116  5.418 

7.114  9.317  12.17 

15.86 

30 

2.427  2.807  3.243 

3.745  4.322  5.743 

7.612  10.06  13.27 

17.45 

31 

2.500  2.905  3.373 

3.914  4.538  6.088 

8.145  10.87  14.46 

19.19 

32 

2.575  3.007  3.508 

4.090  4.765  6.453 

8.715  11.74  15.76 

21.11 

33 

2.652  3.112  3.648 

4.274  5.003  6.841 

9.325  12.68  17.18 

23.23 

34 

2.732  3.221  3.794 

4.466  5.253  7.251 

9.978  13.69  18.73 

25.55 

35 

2.814  3.3.34  3.946 

4.667  5.516  7.686 

10.68  14.79  20.41 

28.10 

36 

2.898  3.450  4.104 

4.877  5.792  8.147 

11.42  15.97  22.25 

30.91 

37 

2.985  3.571  4.268 

5.097  6.081  8.636 

12.22  17.25  24.25 

34.00 

38 

3.075  3.696  4.439 

5.326  6.385  9.154 

13.08  18.63  26.44 

37.40 

39 

3.167  3.825  4.616 

5.566  6.705  9.704 

13.99  20.12  28.82 

41.14 

40 

3.262  3.959  4.801 

5.816  7.040  10.29 

14.97  21.72  31.41 

45.26 

41 

3.360  4.098  4.993 

6.078  7.392  10.90 

16.02  23.46  34.24 

49.79 

42 

3.461  4.241  5.193 

6.352  7.762  11.56 

17.14  25.34  37.32 

54.76 

43 

3.565  4.390  5.400 

6.637  8.150  12.25 

18.34  27.37  40.68 

60.24 

44 

3.671  4.543  5.617 

6.936  8.557  12.99 

19.63  29.56  44.34 

66.26 

45 

3.782  4.702  5.841 

7.248  8.985  13.76 

21.00  31.92  48.33 

72.89 

46 

3.895  4.867  6.07i 

7.574  9.434  14.59 

22.47  34.47  52.68 

80.18 

47 

4.012  5.037  6.318 

7.915  9.906  15.47 

24.05  37.23  57.42 

88.20 

48 

4.132  5.214  6.571 

8.271  10.40  16.39 

25.73  40.21  62.59 

97.02 

49 

4.256  5.396  6.833 

8.644  10.92  17.38 

27.53  43.43  68.22 

106.7 

1000 


1  + 


100  J 


XXIV.    PRESENT   VALUE   OF  1000 
DUE  u  YEARS   HENOE. 


77 


n 

3        3^        4 

4}        5         6 

7         8         9 

10 

0 

1000    1000    1000 

1000    1000    1000 

1000    1000    1000 

1000 

1 
2 
3 

970.9  966.2  961.5 
942.6  933.5  924.6 
915.1  901.9  SS9.0 

956.9  952.4  943.4 
915.7  907.0  890.0 
876.3  863.8  839.6 

934.6  925.9  917.4 
873.4  857.3  841.7 
816.3  793.8  772.2 

909.1 
826.4 
751.3 

4 
5 
6 

855.5  871.4  854.8 

862.6  842.0  821.9 
837.5  813.5  790.3 

838.6  822.7  792.1 
802.5  783.5  747.3 
767.9  746.2  705.0 

762.9  735.0  708.4 
713.0  680.6  649.9 
666.3  630.2  596.3 

683.0 
620.9 
564.5 

7 

8 
9 

813.1  786.0  759.9 
789.4  759.4  730.7 
766.4  733.7  702.6 

734.8  710.7  665.1 
703.2  676.8  627.4 

672.9  644.6  591.9 

622.7  5S3.5  547.0 
582.0  540.3  501.9 
543.9  500.2  460.4 

513.2 
466.5 
424.1 

10 

744.1  708.9  675.6 

643.9  613.9  558.4 

508.3  463.2  422.4 

385.5 

11 
12 
13 

722.4  684.9  649.6 
701.4  661.8  624.6 
681.0  639.4  600.6 

616.2  584.7  526.8 
589.7  556.8  497.0 

564.3  530.3  468.8 

475.1  428.9  387.5 
444.0  397.1  355.5 
415.0  367.7  326.2 

350.5 
318.6 
289.7 

14 
15 
IG 

661.1  617.8  577.5 
641.9  596.9  555.3 

623.2  576.7  533.9 

540.0  505.1  442.3 
516.7  481.0  417.3 
494.5  458.1  393.6 

387.8  340.5  299.2 
362.4  315.2  274.5 
338.7  291.9  251.9 

263.3 
239.4 
217.6 

17 
18 
19 

605.0  557.2  513.4 
587.4  538.4  493.6 
570.3  520.2  474.6 

473.2  436.3  371.4 
452.8  415.5  350.3 

433.3  395.7  330.5 

316.6  270.3  231.1 
295.9  250.2  212.0 
276.5  231.7  194.5 

197.8 
179.9 
163.5 

20 

553.7  502.6  456.4 

414.6  376.9  311.8 

258.4  214.5  178.4 

148.6 

21 

22 
23 

537.5  485.6  438.8 
521.9  469.2  422.0 
506.7  453.3  405.7 

396.8  358.9  294.2 
379.7  341.8  277.5 
363.4  325.6  261.8 

241.5  198.7  163.7 
225.7  183.9  150.2 
210.9  170.3  137.8 

135.1 
122.8 
111.7 

24 
25 
26 

491.9  438.0  390.1 

477.6  423.1  375.1 

463.7  408.8  360.7 

347.7  310.1  247.0 
332.7  295.3  233.0 
318.4  281.2  219.8 

197.1  157.7  126.4 

184.2  146.0  116.0 
172.2  135.2  106.4 

101.5 
92.30  ■ 
83.91 

27 
28 
29 

450.2  395.0  346.8 
437.1  381. 7  333.5 

424.3  368.7  320.7 

304.7  267.8  207.4 
291.6  255.1   195.6 
279.0  242.9  184.6 

160.9  125.2  97.61 
150.4  115.9  89.55 
140.6  107.3  82.15 

76.28 
69.34 
63.04 

30 

412.0  356.3  308.3 

267.0  231.4  174.1 

131.4  99.38  75.37 

57.31 

31 
32 
33 

400.0  344.2  296.5 
388.3  332.6  285.1 
377.0  321.3  274.1 

255.5  220.4  164.3 
244.5  209.9  155.0 
234.0  199.9  146.2 

122.8  92.02  69.15 
114.7  85.20  63.44 
107.2  78.89  58.20 

52.10 
47.36 
43.06 

34 
35 
36 

366.0  310.5  263.6 
355.4  300.0  253.4 
345.0  2S9.8  243.7 

223.9  190.4  137.9 
214.3  181.3  130.1 
205.0  172.7  122.7 

100.2  73.05  53.39 
93.66  67.63  48.99 
87.54  62.62  44.94 

39.14 
35.58 
32.35 

37 

38 
39 

40 

335.0  2S0.0  234.3 
325.2  270.6  225.3 
315.8  261.4  216.6 

196.2  164.4  115.8 
1S7.8  156.6  109.2 
179.7  149.1  103.1 

81.81  57.99  41.23 
76.46  53.69  37.83 
71.46  49.71  34.70 

29.41 
26.73 
24.30 

306.6  252.6  20S.3 

171.9  142.0  97.22 

66.78  46.03  31.84 

22.09 

41 
42 
43 

297.6  244.0  200.3 
289.0  235.8  192.6 
280.5  227.8  185.2 

164.5  135.3  91.72 
157.4  128.8  86.53 
150.7  122.7  81.63 

62.41  42.62  29.21 
58.33  39.46  26.80 
54.51  36.54  24.58 

20.09 
18.26 
16.60 

44 
45 
46 

272.4  220.1  178.0 
264.4  212.7  171.2 
256.7  205.5   164.6 

144.2  116.9  77.01 
138.0  111.3  72.65 
132.0  106.0  68.54 

50.95  33.83  22.55 
47.61  31.33  20.69 
44.50  29.01  18.98 

15.09 
13.72 
12.47 

47 

48 
49 

249.3  198.5  158.3 
242.0  191.8  152.2 
235.0  185.3  146.3 

126.3  100.9  64.66 
120.9  96.14  61.00 
115.7  91.56  57.55 

41.59  26.86  17.42 
38.87  24.87  15.98 
36.32  23.03  14.66 

11.34 
10.31 
9.370 

78 


XXV.  AMOUNT  OF  AN  ANNUITY  PAID 
AT  THE  END  OP  EAOH  YEAR. 


i'^mj 


100 


n 

3       3i        4 

4!,        6       6 

7          8 

9 

10 

0 

0.000  0.000  0.000 

0.000  0.000  0.000 

0.000  0.000  0.000 

0.000 

1 
2 

3 

1.000  1.000  1.000 
2.030  2.035  2.040 
3.091  3.106  3.122 

1.000  1.000  1.000 
2.045  2.050  2.060 
3.137  3.153  3.184 

1.000  1.000 
2.070  2.080 
3.215  3.246 

1.000 
2.090 
3.278 

1.000 
2.100 
3.310 

4 
5 

G 

4.1S4  4.215  4.246 
,S.309  5.362  5.416 
6.468  6.550  6.633 

4.278  4.310  4.37i 
5.471  5.526  5.637 
6.717  6.802  6.975 

4.440  4.506  4.573 
5.751  5.867  5.985 
7.153  7.336  7.523 

4.641 
6.105 
7.716 

7 
8 
9 

7.662  7.779  7.S9S 
S.S92  9.052  9.214 
10.16  10.37  10.58 

8019  8.142  8.394 
9.380  9.549  9.897 
10.80  11.03  11.49 

8.654  8.923  9.200 
10.26  10.64  11.03 
11.98  12.49  13.02 

9.487 
11.44 
13.58 

10 

11.46  11.73  12.01 

12.29  12.58  13.18 

13.82  14.49 

15.19 

15.94 

11 
12 
13 

12.81  13.14  13.49 
14.19  14.60  15.03 
15.62  16.11   16.63 

13.84  14.21   14.97 
15.46  15.92  16.87 
17.16  17.71  18.88 

15.78  16.65 
17.89  18.98 
20.14  21.50 

17.56 
20.14 
22.95 

18.53 
21.38 
24.52 

U 
15 
16 

17.09  17.68  18.29 
18.60  19.30  20.02 
20.16  20.97  21.82 

18.93  19.60  21.02 
20.78  21.58  23.28 
22.72  23.66  25.67 

22.55  24.21 
25.13  27.15 
27.89  30.32 

26.02 
29.36 
33.00 

27.97 
31.77 
35.95 

17 

18 
19 

21.76  22.71  23.70 
23.41  24.50  25.65 
25.12  26.36  27.67 

24.74  25.84  28.21 
26.86  28.13  30.91 
29.06  30.54  33.76 

30.84  33.75 
34.00  37.45 
37.38  41.45 

36.97 
41.30 
46.02 

40.54 
45.60 
51.16 

57.27 

20 

26.87  28.28  29.78 

31.37  33.07  36.79 

41.00  45.76 

51.16 

21 
22 
23 

28.68  30.27  31.97 
30.54  32.33  34.25 
32.45  34.46  36.62 

33.78  35.72  39.99 
36.30  38.51  43.39 
38.94  41.43  47.00 

44.87  50.42  56.76 
49.01  55.46  62.87 
53.44  60.89  69.53 

64.00 
71.40 
79.54 

24 
25 
26 

34.43  36.67  39.08 
36.46  38.95  41.65 
38.55  41.31  44.31 

41.69  44.50  50.82 
44.57  47.73  54.86 
47.57  51.11  59.16 

58.18  66.76 
63.25  73.11 
68.68  79.95 

76.79 
84.70 
93.32 

88.50 
98.35 
109.2 

27 

28 
29 

40.71  43.76  47.08 
42.93  46.29  49.97 
45.22  48.91  52.97 

50.71  54.67  63.71 
53.99  58.40  68.53 
57.42  62.32  73.64 

74.48  87.35 
80.70  95.34 
87.35   104.0 

102.7 
113.0 
124.1 

121.1 
134.2 
148.6 

30 

47.58  51.62  56.08 

61.01  66.44  79.06 

94.46  113.3 

136.3 

164.5 

31 
32 
33 

50.00  54.43  59.33 
52.50  57.33  62.70 
55.08  60.34  66.21 

64.75  70.76  84.80 
68.67  75.30  90.89 

72.76  80.06  97.34 

102.1  123.3 

110.2  134.2 
118.9  146.0 

149.6 
164.0 
179.8 

181.9 
201.1 
222.3 

34 
35 
36 

57.73  63.45  69.86 
60.46  66.67  73.65 
63.28  70.01  77.60 

77.03  85.07  104.2 
81..S0  90.32  111.4 
86.16  95.84  119.1 

128.3   158.6 
138.2  172.3 
148.9  187.1 

197.0 
215.7 
236.1 

245.5 
271.0 
299.1 

37 
38 
39 

66.17  73.46  81.70 
69.16  77.03  85.97 
72.23  80.72  90.41 

91.04  101.6  127.3 
96.14  107.7  135.9 
101.5  114.1  145.1 

160.3  203.1 
172.6  220.3 
185.6  238.9 

258.4 
282.6 
309.1 

330.0 
364.0 
401.4 

40 

75.40  84.55  95.03 

107.0  120.8  154.8 

199.6  259.1 

337.9 

442.6 

41 
42 
43 

78.66  88.51  99.83 
82.02  92.61  104.8 
85.48  96.85  110.0 

112.8  127.8  165.0 

118.9  135.2  176.0 
125.3  143.0  187.5 

214.6  280.8  369.3 
230.6  304.2  403.5 
247.8  329.6  440.8 

487.9 
537.6 
592.4 

44 
45 
46 

89.05  101.2  115.4 
92.72  105.8  121.0 
96..S0  110.5  126.9 

131.9  151.1  199.8 
138.8  1.59.7  212.7 
146.1  168.7  226.5 

266.1  356.9 

285.7  386.5 

306.8  418.4 

481.5 
525.9 
574.2 

652.6 
718.9 
791.8 

47 
4^ 

4!) 

100.4  115.4  132.9 
104.4  120.4  139.3 
108  5  125.6  145.8 

1.53.7  178.1  241.1 
161.6  188.0  256.6 
169.9  1984  273.0 

329.2  452.9  626.9 

353.3  490.1  684.3 
379.0  530.3  746.9 

872.0 
960.2 
1057 

(i  +  ^Y 
V     loo; 


100 ' 


/+4)' 


XXVL  PRESENT  VALUE  OF  AN  ANNU- 
ITY PAID  AT  THE  END  OP  EACH 
YEAR. 


n 

3        3i        4 

4,V        5        6 

7          8         9 

10 

0 

0.000  0.000  0.000 

0.000  0.000  0.000 

0.000  0.000  0.000 

0.000 

1 
2 
3 

.9709  .9662  .9615 
1.913  1.900  1.8S6 
2.829  2.802  2.775 

.9569  .9524  .9434 
1.873  1.859  1.833 
2.749  2.723  2.673 

.9346  .9259  .9174 
1.808  1.783  1.759 
2.624  2.577  2.531 

.9091 
1.736 

2.487 

4 
5 
6 

3.717  3.673  3.630 
4.580  4.515  4.452 
5.417  5.329  5.242 

3.5SS  3.546  3.465 
4.390  4.329  4.212 
5.158  5.076  4.917 

3.387  3.312  3.240 
4.100  3.993  3.890 
4.767  4.623  4.486 

3.170 
3.791 
4.355 

7 
8 
9 

6.230  6.115  6.002 
7.020  6.874  6.733 
7.786  7.608  7.435 

5.893  5.7S6  5.582 
6.596  6.463  6.210 
7.269  7.108  6.802 

5.389  5.206  5.033 
5.971  5.747  5.535 
6.515  6.247  5.995 

4.868 
5.335 
5.759 

10 

8.530  8.317  8.111 

7.913  7.722  7.360 

7.024  6.710  6.418 

6.145 

11 
12 
13 

9.253  9.002  8.760 
9.954  9.663  9.385 
10.63  10.30  9.986 

8.529  8.306  7.887 
9.119  8.863  8.384 
9.683  9.394  8.853 

7.499  7.139  6.805 
7.943  7.536  7.161 
8.358  7.904  7.487 

6.495 
6.814 
7.103 

14 
15 
16 

11.30  10.92  10.56 
11.94  11.52  11.12 
12.56  12.09  11.65 

10.22  9.899  9.295 
10.74  10.38  9.712 

11.23  10.84  10.11 

8.745  8.244  7.786 
9.108  8.559  8.061 
9.447  8.851  8.313 

7.367 
7.606 
7.824 

17 

18 
19 

13.17  12.65  12.17 
13.75  13.19  12.66 
14.32  13.71  13.13 

11.71  11.27  10.48 
12.16  11.69  10.83 
12.59  12.09  11.16 

9.763  9.122  8.544 
10.06  9.372  8.756 
10.34  9.6(H  8.950 

8.022 
8.201 
8.365 

20 

14.88  14.21  13.59 

13.01  12.46  11.47 

10.59  9.818  9.129 

8.514 

21 
22 
23 

15.42  14.70  14.03 
15.94  15.17  14.45 
16.44  15.62  14.86 

13.40  12.82  11.76 
13.78  13.16  12.04 
14.15  13.49  12.30 

10.84  10.02  9.292 
11.06  10.20  9.442 
11.27  10.37  9.580 

8.649 
8.772 
8.883 

24 
25 
26 

16.94  16.06  15.25 
17.41  16.48  15.62 
17.88  16.89  15.98 

14..50  13.80  12.55 
14.83  14.09  12.78 
15.15  14.38  13.00 

11.47  10.53  9.707 
11.65  10.67  9.823 
11.83  10.81  9.929 

8.985 
9.077 
9.161 

27 

28 
29 

30 

18.33  17.29  16.33 
18.76  17.67  16.66 
19.19  18.04  16.98 

15.45  14.64  13.21 
15.74  14.90  13.41 
16.02  15.14  13.59 

11.99  10.94  10.03 
12.14  11.05  10.12 
12.28  11.16  10.20 

9.237 
9.307 
9.370 

19.60  18.39  17.29 

16.29  15.37  13.76 

12.41  11.26  10.27 

9.427 

31 
32 
33 

20.00  18.74  17.59 
20.39  19.07  17.87 
20.77  19.39  18.15 

16.54  15.59  13.93 
16.79  15.80  14.08 
17.02  16.00  14.23 

12..S3  11.35  10.34 
12.65  11.43  10.41 
12.75  11.51  10.46 

9.479 
9.526 
9.569 

34 
35 
36 

21.13  19.70  18.41 
21.49  20.00  18.66 
21.83  20.29  18.91 

17.25  16.19  14.37 
17.46  16.37  14.50 
17.67  16.55  14.62 

12.85  11.59  10.52 
12.95  11.65  10.57 
13.04  11.72  10.61 

9.609 
9.644 
9.677 

37 
38 

39 

22.17  20.57  19.14 
22.49  20.84  19.37 
22.81  21.10  19.58 

17.86  16.71  14.74 
18.05  16.87  14.85 
18.23  17.02  14.95 

13.12  11.78  10.65 
13.19  11.83  10.69 
13.26  11.88  10.73 

9.706 
9.733 
9.757 

40 

23.11  21.36  19.79 

18.40  17.16  15.05 

13.33  11.92  10.76  :  9.779   | 

41 
42 
43 

23.41  21.60  19.99 
23.70  21.83  20.19 
23.98  22.06  20.37 

18.57  17.29  15.14 
18.72  17.42  15.22 
18.87  17.55  15.31 

13.39  11.97  10.79 
13.45  12.01  10.81 
13.51  12.04  10.84 

9.799 
9.817 
9.834 

44 
45 
46 

24.25  22.28  20.55 
24.52  22.50  20.72 
24.78  22.70  20.88 

19.02  17.66  15.38 
19.16  17.77  15.46 
19.29  17.88  15.52 

13.56  12.08  10.86 
13.61  12.11  10.88 
13.65  12.14  10.90 

9.849 
9.863 
9.875 

47 

48 
49 

25.02  22.90  21.04 
25.27  23.09  21.20 
25.50  23.28  21.34 

19.41  17.98  15.59 
19..S4  18.08  15.65 
19.65  18.17  15.71 

13.69  12.16  10.92 
13.73  12.19  10.93 
13.77  12.21  10.95 

9.887 
9.897 
9.906 

80  XXVII.  AMOUNT  OF  AN  AN- 
NUITY WHEN  PAID  AT  THE 
BEGINNING  OF  EACH  YEAR. 


1  + 


1  + 


100  y 


100 


n 

8       3^        4 

4,V        5        6 

7          8          9 

10 

0 

1.000  1.000  1.000 

1.000  1.000  1.000 

1.000  1.000  1.000 

1.000 

1 
2 
3 

1.030  1.035  1.040 
2.0*;!  2.106  2.122 
3.184  3.215  3.246 

1.045  1.050  1.060 
2.137  2.153  2.184 
3.278  3.310  3.375 

1.070  1.080  1.090 
2.215  2.246  2.278 
3.440  3.506  3.573 

1.100 
2.310 
3.641 

4 
6 

6 

4.309  4.362  4.416 
5.468  5.5.50  5.633 
6.662  6.779  6.898 

4.471  4.526  4.637 
5.717  5.802  5.975 
7  019  7.142  7.394 

4.751  4.867  4.985 
6.153  6.3.36  6.523 
7.654  7.923  8.200 

5.105 
6.716 
8.487 

7 
8 
9 

7.S92  S0S2  8.214 
9.159  9..368  9.583 
10.46  10.73  11.01 

8.380  8.549  8.897 
9.802  10.03  10.49 
11.29  11  58  12.18 

9.260  9.637  10.03 
10.98  11.49  12.02 
12.82  13.49  14.19 

10.44 
12.58 
14.94 

10 

11.81  12.14  12.49 

12.84  13.21  13.97 

14.78  15.65  16.56 

17.53 

11 
12 
13 

13.19  13.60  14.03 
14.62  15.11  15.63 
16.09  16.68  17  29 

14.46  14.92  15.87 
16.16  16.71   17.88 
17.93  18.60  2O02 

16.89  17.98  19.14 
19.14  20.50  21.95 
21.55  23.21  25.02 

20.38 
23.52 
26.97 

14 
15 
16 

17.60  18.30  19.02 
19.16  19.97  2082 
20.76  21.71  22.70 

19.78  20.58  22.28 
21.72  22.66  24.67 
23.74  24.84  27  21 

24.13  26.15  28.36 
26.89  29.32  32.00 
29.84  32.75  35.97 

30.77 
34.95 
39.54 

17 

18 
19 

22.41  23..SO  24  65 
24.12  25.36  26.67 
25.87  27.28  28.78 

25.86  27.13  29.91 
28.06  29.54  32.76 
30.37  32.07  35.79 

33.00  36.45  40.30 
36.38  40.45  45.02 
40.00  44.76  5016 

44.60 
50.16 
56.27 

20 

27.68  29.27  30.97 

32.78  34.72  38.99 

43.87  49.42  55.76 

63.00 

21 
22 
23 

29.54  31.33  33.25 
31.45  33.46  35.62 
33.43  35.67  38.08 

35.30  37.51  42,39 
37.94  40.43  46.00 
40.69  43.50  49.82 

48.01  54.46  61.87 
52.44  59.89  68.53 
57.18  65.76  75.79 

70.40 

78.54 
87.50 

24 
25 
26 

35.46  37.95  40.65 
37.55  40.31  43.31 
39.71  42.76  46.08 

43.57  46.73  53.86 
46.57  50.11  58.16 
49.71  53.67  62.71 

62.25  72.11  83.70 
67.68  78.95  92.32 
73.48  86.35  101.7 

97.35 
108.2 
120.1 

27 

28 
29 

41.93  45.29  48.97 
44.22  47.91  51.97 
46.58  50.62  55.08 

52.99  57.40  67.53 
56.42  61.32  72.64 
60.01  65.44  78.06 

79.70  94.34  112.0 
86.35  103.0  123.1 
93.46  112.3  135.3 

133.2 
147.6 
163.5 

30 

49.00  53.43  58.33 

63.75  69.76  83.30 

101. 1  122.3  148.6 

180.9 

31 
32 
33 

51.50  56.33  61.70 
54.08  59.34  65.21 
56.73  62.45  6S.S6 

67.67  74.30  89.89 
71.76  79.06  96.34 
76.03  84.07  103.2 

109.2  133.2  163.0 
117.9  145.0  17S.S 

127.3  157.6  196.0 

200.1 
221.3 
244.5 

34 
35 
36 

.5'9.46  65.67  72.65 
62.28  69.01   76.60 
65.17  72.46  80.70 

80.50  89.32  110.4 
85.16  94.84  118.1 
90.04  100.6  126.3 

137.2  171.3  214.7 
147.9  186.1  235.1 

159.3  202.1  257.4 

270.0 
298.1 
329.0 

37 

38 
39 

40 

68.16  76.03  84.97 
71.23  79.72  89.41 
74.40  83.55  94.03 

95.14  106.7  134.9 
100.5  113.1  144.1 
106.0  119.8  153.8 

171.6  219.3  281.6 
184.6  237.9  308.1 
198.6  258.1  336.9 

363.0 
400.4 
441.6 

77.66  87.51  98.83 

111.8  126.8  164.0 

213.6  279.8  368.3 

486.9 

41 
42 
43 

81.02  91.61  103.8 
84.48  95.85  109.0 
88.05  100.2  114.4 

117.9  134.2  175.0 
124.3  142.0  186.5 
130.9  150.1  198.8 

229.6  303.2  402.5 
246.8  328.6  439.8 
265.1  355.9  480.5 

536.6 
591.4 
651.6 

44 
45 
46 

91.72  104.8  1200 
95  50  109.5  125.9 
99.40  114.4  131.9 

137.8  158.7  211.7 
145.1  167.7  225.5 
152.7  177.1  240.1 

284.7  385.5  524.9 

305.8  417.4  573.2 
328.2  451.9  625.9 

717.9 
790.8 
871.0 

47 
4S 

103.^  119.4  138.3 
107.5  124.6  144.8 
Ills  1,300  151.7 

160.6  187.0  255.6 
168.9  197.4  272.0 
177.5  208.3  289.3 

352.3  489.1  683.3 
378.0  529.3  745.9 
405.5  572.8  814.1 

959.2 
1056 
1163 

iooo(i  +  -^Yj:~ 

[        100  J    100 


1  + 


looy 


XXVIII.  SUM  TO  BE  PAID  AT 
THE  END  or  EACH  OF  u  YEAES 
TO  EXTINGUISH  A  DEBT  OP  1000. 


81 


n 

3       3^       4 

4i        5        6 

7         8         9 

10 

1 
2 

3 

1030    1035    1040 
522.6  526.4  530.2 
353.5  356.9  360.3 

1045    1050    1060 
534.0  537.8  545.4 
363.8  367.2  374.1 

1070    1080    1090 
553.1  560.8  568.5 
381.1  388.0  395.1 

1100 
576.2 
402.1 

4 
5 
6 

269.0  272.3  275.5 
218.4  221.5  224.6 
184.6  187.7  190.8 

278.7  2S2.0  288.6 

227.8  231.0  237.4 

193.9  197.0  203.4 

295.2  301.9  308.7 
243.9  250.5  257.1 
209.8  216.3  222.9 

315.5 
263.8 
229.6 

7 
8 
9 

160.5  163.5  166.6 
142.5  145.5  148.5 
128.4  131.4  134.5 

169.7  172.8  179.1 
151.6  154.7  161.0 
137.6  140.7  147.0 

185.6  192.1  198.7 
167.5  174.0  180.7 
153.5  160.1  166.8 

205.4 
187.4 
173.6 

10 

117.2  120.2  123.3 

126.4  129.5  135.9 

142.4  149.0  155.8 

162.7 

11 
12 
13 

108.1  111.1  114.1 
100.5  103.5  106.6 
94.03  97.06  100.1 

117.2  120.4  126.8 
109.7  112.8  119.3 

103.3  106.5  113.0 

133.4  140.1  146.9 
125.9  132.7  139.7 
119.7  126.5  133.6 

154.0 
146.8 
140.8 

14 
15 
16 

88.53  91.57  94.67 
83.77  86.83  89.94 
79.61  82.68  85.82 

97.82  101.0  107.6 
93.11  96.34  103.0 
89.02  92.27  98.95 

114.3  121.3  128.4 

109.8  116.8  124.1 

105.9  113.0  120.3 

135.7 
131.5 
127.8 

17 

18 
19 

75.95  79.04  82.20 
72.71  75.82  78.99 
69.81  72.94  76.14 

85.42  88.70  95.44 
82.24  85.55  92.36 
79.41  82.75  89.62 

102.4  109.6  117.0 
99.41  106.7  114.2 
96.75  104.1  111.7 

124.7 
121.9 
119.5 

20 

67.22  70.36  73.58 

76.88  80.24  87.18 

94.39  101.9  109.5 

117.5 

21 
22 
23 

64.87  68.04  71.28 
62.75  65.93  69.20 
60.81  64.02  67.31 

74.60  78.00  85.00 
72.55  75.97  83.05 
70.68  74.14  81.28 

92.29  99.83  107.6 
90.41  98.03  105.9 
88.71  96.42  104.4 

115.6 
114.0 
112.6 

24 
25 
26 

59.05  62.27  65.59 
57.43  60.67  64.01 
55.94  59.21  62.57 

68.99  72.47  79.68 
67.44  70.95  78.23 
66.02  69.56  76.90 

87.19  94.98  103.0 
85.81  93.68  101.8 
84.56  92.51  100.7 

111.3 
110.2 
109.2 

27 

28 
29 

30 

54.56  57.85  61.24 
53.29  56.60  60.01 
52.11  55.45  58.88 

64.72  68.29  75.70 
63.52  67.12  74.59 
62.41  66.05  73.58 

83.43  91.45  99.73 
82.39  90.49  98.85 
81.45  89.62  98.06 

108.3 
107.5 
106.7 

51.02  54.37  57.83 

61.39  65.05  72.65 

80.59  88.83  97.34 

106.1 

31 
32 
33 

50.00  53.37  56.86 
49.05  52.44  55.95 
48.16  51.57  55.10 

60.44  64.13  71.79 
59.56  63.28  71.00 
58.74  62.49  70.27 

79.80  88.11  96.69 
79.07  87.45  96.10 
78.41  86.85  95.56 

105.5 
105.0 
104.5 

34 
35 
36 

47.32  50.76  54.31 
46.54  50.00  53.58 
45.80  49.28  52.89 

57.98  61.76  69.60 
57.27  61.07  68.97 
56.61  60.43  68.39 

77.80  86.30  95.08 
77.23  85.80  94.64 
76.72  85.34  94.24 

104.1 
103.7 
103.3 

37 

38 
39 

45.11  48.61  52.24 
44.46  47.98  51.63 
43.84  47.39  51.06 

55.98  59.84  67.86 
55.40  59.28  67.36 
54.86  58.76  66.89 

76.24  84.92  93.87 
75.80  84.54  93.54 
75.39  84.19  93.24 

103.0 
102.7 
102.5 

40 

43.26  46.83  50.52 

54.34  58.28  66.46 

75.01  83.86  92.96 

102.3 

41 
42 
43 

42.71  46.30  50.02 
42.19  45.80  49.54 
41.70  45.33  49.09 

53.86  57.82  66.06 
53.41  57.39  65.68 
52.98  56.99  65.33 

74.66  83.56  92.71 
74.34  83.29  92.48 
74.04  83.03  92.27 

102.0 
101.9 
101.7 

44 
45 

46 

41.23  44.88  48.66 
40.79  44.45  48.26 
40.36  44.05  47.88 

52.58  56.62  65.01 
52.20  56.26  64.70 
51.84  55.93  64.41 

73.76  8280  92.08 
73.50  82.59  91.90 
73.26  82.39  91.74 

101.5 
101.4 
101.3 

47 

48 
49 

39.96  43.67  47.52 
39.58  43.31  47.18 
39.21  42.96  46.86 

51.51  55.61  64.15     73.04  82.21  91.60 
51.19  55.32  63.90    72.83  82.04  91.46 
50.89  55.04  63.66    7264  81.89  91.34 

101.1 
101.0 
100.9 

XXIX.     LEAST  DIVISOES. 


0 

1 

V 

3 

4 

n 

1 

3 

7    9 

1 

8 

7 

9 

I 

3 

7 

9 

1 

3    7 

9 

13    7    9 

0 





-    3 









3 



3 





3  — 

3 

7 

1 
2 

3 

3 
7 

7 
3 

3  11 
—    3 

3 

3 

3 

7 

7 
3 
11 

11 
13 
3 

3 
17 

3 

3 

7 

3 

7  — 

-    3 

3  - 

3 

3  11     3  - 
-    3  13     3 
11     7 

4 
5 
6 

3 

13 
3 

11  — 

3  - 

—    3 

3 

7 
13 

7 
3 

3 
11 

3 

1 

3 

7 

7    3 

17  23 
3  17 

3 

—  19 
13     3 

3     7 

7 
3 

3-    .3- 
-    3  -    3 
11 

7 
8 
9 

~3 
17 

19 
11 
3 

7  — 

3  - 

—    3 

3  23 

—  3 

—  11 

3 
19 

7 

3 

7 
3 

3 
13 

3 

3 

17 
3 
7 

—  11 
7    3 
3  — 

3 

3—37 
29    3     7     3 
—  23  —  13 

10 

7 

17 

19  - 

3 

— 

3 

— 

— 

3 

13 

3 

— 

-  17 

— 

3     7    3  — 

11 
12 
13 

3 

3 

3  — 

17    3 

—     7 

11 

7 
3 

3 

13 

3 

3 
23 

19 
3 

3 

7 
3 

- 
3 

3 

11 

11     3 
3  — 

31     7 

17 
3 
13 

7    3  31     3 
17  11  29  - 
3  17    3  19 

14 
15 
16 

3 
19 

23 
3 

7 

3  - 
11     3 

17 
3 

3  13 
17  37 
—    3 

3 

7 

7 
3 

3 

3 

", 

3—3 

—    3  29 

7  23  — 

3 
11 

11     3  —    3 

23  —     7  — 

3  31     3  17 

17 

18 
19 

3 

13 
3 
11 

3  — 
13    3 
-  23 

29 
"3 

3  17 

7  23 

-    3 

3 
17 
19 

3 

17 

—  11     7 

-  3  31 
3  41     3 

3 

—  3  37 
3  11    3 

—  13    7 

—    3-3 
7  19  -  43 
3  29    3  — 

20 

3 

— 

3     7 

— 

3 

— 

3 

43 

7 

— 

— 

3 

19    3 

— 

13    3  23    3 

21 
23 

11 

31 

3 

3 

7 

7    3 

—  47 

3  — 

J 

3 

29 
3 

7 

13 

7 
3 

3 
11 

11 

3 

23 

3 

17 
13 

3 
17 

23 
3 

3  — 

7  — 
—    3 

3 

19    7 

3  —    3  13 
-3—3 

24 
25 
26 

7 

41 

3 

3 

19 

29    3 

23  13 

3  — 

3 

7 

19 

7 
3 

3 

41 
11 
3 

3 

—    3 

3     7 
43  37 

7 
3 
11 

11 
3 

3  - 

17  43 
—    3 

3 

7 

—    7  —  31 
3  -    3  - 
19    3  —    3 

27 

28 
29 

37 
3 

3 

—    3 
7  53 
3  - 

3  29 
41     3 

11 
3 

3 

3     7 

7    3 

23  37 

3 
11 

3 
29 

19 
3 

3     7 
7    3 

3 

17 

—  13  41  - 
3-37 
17    3     7     3 

30 

— 

3  31    3 

— 

23 

7 

— 

3 

— 

3 

13 

7 

3  — 

3 

-  17  11  — 

31 
32 
33 

7  29  13  — 

3  —    3  — 

—    3  —    3 

3 

13 

7 

11 
3 

3 
31 

3 

3 

3 

11 

53 
7 
3 

3 

31 
3 

13  —  43 

53    3  41 

3  47    3 

3     7    3  47 

7    3  17    3 

13 17 

34 
35 
36 

19  41 
3  31 
13     3 

—  7 
3  11 

—  3 

3 
23 

3 

3 

13 
3 

7 

11 
7 
3 

3 
13 

23 
3 

3 
19 

i 

-    3 
3  - 

19 
3 

3  11     3  - 
-3-3 
11  —    7  41 

37 
38 
39 

40^ 

3 
47 

7 
3 

11  - 

3  13 

—    3 

3  47 
37     3 

-    7 

3 

11 

3 

61 
3 

3 

43 
3 

3 

7 

7 
3 

—  37 

—  3 
3  31 

11 
3 

3  19    3  23 
23     3  —    3 

7 11 

— 

— 

-  19 

3 

_ 

3 

— 

— 

3 

— 

3    29  37  11 

7 

3  13    3  - 

41 
42 
43 

3 
11 

11 

3 

13 

3     7 

7    3 

59  31 

3 

3 
11 

19 

23 
3 

3 

7 

13 

3 

29 

7 

41 

3 

3 

3 

3 
61 

—    3 
3  19 

7  — 

3 

41     3  11    3 

31     7 

3  43     3  — 

44 
45 
46 

3 

7 
43 

7 
3 

3  — 
—    3 
17  11 

11 
13 
3 

3 

7 

7    3 
3  31 

3 

3 

19  43 
3     7 

7    3 

3 

23 

11 

11     3 
3  13 

41  - 

23 
3 

—    3-3 

19    7 

3  -    3  - 

47 

4S 
49 

3 

13 

3 

3  17 
11     3 

7  — 

7 
17 
3 

3 

17 

53 
3 

3 
61 

3 

7 

7 
3 

29 
3 
13 

11 
3 

3 

—    3 

3     7 

3 
11 

11     3  47    3 

47  29  37  13 

3-37 

XXIX.    LEAST  DIVISOES. 


5 

« 

7 

8 

9 

u 

1 

3 

7 

9 

1 

3 

7  9 

1  3  7 

9 

1 

3 

7 

9 

1 

3  7  9 

0 

3 

— 

3 

— 

— 

3 

-  3 

7 

— 

3 



3 



7 

3-3 

1 
2 
3 

3 

3 
11 

3 

3 

7 

7 
3 
19 

3 

-  13 
3  — 

-  3 

3  —  3 

—  3  — 

7  —  13 

3 

3 

3 

11 
7 
3 

3 

17 

3 

17 

—  3  13 
3—3 

4 
5 
6 

11 
19 
3 

3 

7 

3 

3 
13 

3 

3  23  3 

3  11  3 

11 

~3 

7 

13 

7 
3 

3 
11 

3 

3 

19 
13 

3 

17  7  - 

-  3  - 

3  17  3 

7 
8 
9 

23 
3 

3 

3 

3 

7 

3 
31 

7 
3 

13  - 

3  11 

—  3 

3—3 
13  3  - 

—  7  — 

19 
3 
11 

11 
3 

3 

-  3 

—  7 
3  23 

7 
3 

13  —  17 
19  3  29 
3-3 

10 

— 

3 

7 

3 

— 

_ 

11  — 

3  29  3 

13 

23 

3 

— 

3 

— 

7 

11 
12 
13 

1 

7 

7 
3 

13 

3 

23 

19 
3 

3 
13 

3 
29 

3  7 

7  3 

—  37 

—  3  11 

31  19  — 

3—3 

3 

7 

1 

7 
3 

3 
19 

29 

1 

3 
13 

-  3  11 
3  —  3 

7  11  — 

14 
15 
16 

3 
13 

3 

31 
3 

3 

3 

7 

11 

7 
3 

3  13 
-  3 

-  3  7 

—  11  19 
3  7  3 

3 
23 

3 
41 

3 

3 

7 

7 
3 

3 

37 
19 

3—3 

17 

18 
19 

17 
3 

17 
3 

7 
3 
19 

11 
3 

3  41 
—  3 
37  13 

3  29 

—  3 

7  11 

7  3  — 
3-3 

3 

_ 

13 
3 

7 

7 
3 

3 

3 

3 
31 
11 

11  3  7 
3  7  3 

20 

7 



11 

29 

3 



3  — 

19  3  31 

3 

— 

— 

— 

— 

3 

7  3  - 

21 
22 
23 

3 

—  3 
3  37 
13  — 

17 
3 

7 

3  17 

11  3 
3  23 

13  41  7  - 

3  -  3  43 

-  3  —  3 

3  37 
—  3 

3 

7 

11 
3 

7 

29 

3 

3  13  3 

11 

—  3  — 

24 
25 
26 

3 
11 

11 
3 

7 

3 

3 

23 
13 
3 

3 
11 

—  3 
17  7 
3  17 

3  31  3 

37 
3 

3 
29 

7 

13 
3 

3 
13 

19 
3 

47 
3 

3  11  3 

-  7  23 

—  3  - 

27 

28 
29 

3 

13 

3 

3 

31 
3 
11 

11 
3 

3—3 

7  47  19 

17  47  — 

3  13  3 

—  3  13 

7 
3 

3 
43 
11 

11  3 
3  — 
19  29 

3 

7 

—  3  —  3 
7  11  —  13 
3  41  3  — 

30 

3  43 

3 

7 



3 

—  3 

37  7  17 

— 

3 

— 

3 

11 

3  19  3 

31 
32 
33 

23 
3 

3 

7 

7 
3 

3 

29 
3 

13 
3 

3  7 
7  3 

3  19  3 
-  3  29 
11 

11 

3 

31 

17 
3 

3 

7 
17 

19 
3 

3 
11 

3 

31  23  7 

37  3  — 

3  43  3 

34 
35 
36 

7 
53 
3 

3 
11 
13 

3 

3 

3 

7 

7 
3 

3  43 
19  3 

3  23  3 

—  3  7 

7 
3 
13 

59 
3 

3 

29 

11  3 
17  37 
3  7 

3 

7  13  — 

—  3  59 

3-3 

37 

38 
39 

11 
3 

3 

59 

13  3 

7  17 
3  37 

3 

17 

S3 
3 

3  53 
-  3 

3  7  3 

7  3  — 

11  29  41 

3 
23 

19 
3 

3 
11 

7 

7 

13 
3 

3 

17 
3 
13 

29 

17  3  7 
3  7  3 

40 

— 

3 

— 

3 

31 

17 

7  13 

3—3 

— 

7 

3 

61 

3 

— 

-  17  - 

41 
42 
43 

7 
3 
19 

3 

3 

3 

3 

7 

23 
3 

3  11 
17  3 
11  17 

43  3  — 
3—3 

3 
11 
29 

37  47  53 
3—3 
13  3  41 

59 
"3 

3 

7 

7  3  13 

3-3 

23  -  53 

44 
45 
46 

3 

61 

29 

3 

—  7 
3  47 

—  3 

3 
59 

3 

3  41 
—  3 

13  7 

17  3  11 
7  17  23 
3—3 

3 
19 

3 
31 

3 

7  67 

3  13 

43  3 

3 

—  3  11 
3—3 
13  7  37 

47 

48 
49 

-  7  67  — 
3  23  3  43 

—  3—3 

3 
11 

11 
3 

7 

3  19 
31  3 

13  3  17 

—  11  — 

3-3 

3 
7 
13 

7 
3 
17 

19 
3 

3 

3 

3 

67 

7 

-  3  - 
3  59  3 

—  19  — 

84 


XXIX.     LEAST   DIVISORS. 


0                     1 

2 

3 

4 

u 

1    3    7    9  1  1    3    7    9 

13    7    9 

13    7    9 

13    7    9 

60 

3  —    3  — 

—    3  29    3 

11  47 

3    7    3- 

71    3    7    3 

51 
52 
58 

—    3  -    3 
7  11  41  - 
3  -    3  — 

19  —    7  - 

3  13    3  17 

47    3  13    3 

3  47    3  23 
23    3  —    3 
17  —    7  73 

7    3  11    3 

13 

3  -    3  19 

53  37  —  19 
3     7    3  29 
7    3—3 

54 
55 
56 

11    3  —    3 
~3  13  1  71 

3  37  1 - 
31     3  41     3 

3  11     3  61 

-    3  -    3 

7  —  17  13 

-  3  -    3 

—  11     7  29 
3  43    3  — 

13  — 

3  23    3  31 
-    3  -    3 

57 

58 
59 

-  3  13    3 

-  7  -  37 
3  —    3  19 

-  29  -     7 

3  —    3  11 

23    3  61     3 

3  59    3  17 
—    3—3 
31 7 

11     3  -    3 
7  19  13  - 
3  17    3  - 

3  —    3  — 
13     3  19    3 

60 

17    3  —    3 

—    7  11  13 

3  19    3  — 

37    3  -    3 

7 23 

61 
62 
63 

—  17  31  41 

3—37 

-373 

3  —    3  29 

—  3-3 

-  59  —  71 

-  3  11     3 

-  7  13  - 
3  -    3  - 

17     7 

3  23    3  17 
13    3  -    3 

3  —    3  11 
79    3  -    3 
17—11     7 

64 
65 
66 

37  19  43  13 
3     7.  3  23 
7    3—3 

3  11    3     7 
17    3     7    3 
11  17  13  — 

—  3  —    3 

—  11  61  — 
3  37    3     7 

59    7  41  47 
3  47    3  13 
19    3  -    3 

3  17    3  — 
31    3  -    3 
29    7  17  61 

67 
68 
69 

19  - 

3  -    3  11 
67    3  -    3 

3     7    3- 

7    3  17    3 

-  31  -  11 

11     3     7    3 

19 

3     7    3  13 

53 23 

3—37 
29    3     7    3 

3  11     3  17 
—    3  41    3 
11  53 

70 

—  47    7  43 

3  -    3  - 

7    3-3 

79  13  31  — 

3-37 

71 
72 
73 

3  —    3  — 
19    3  -    3 
7  67 

13    3  11     3 
3  71     3  13 

-  17 

3  31     3  - 

-  3  17    3 

3     7    3  11 

7    3—3 

11  41 

37    3     7    3 

13 11 

3     7    3- 

74 
75 
76 

3  11    3  31 
13    3  -    3 
11 7 

-3-3 
7  11  -  73 
3  23    3  19 

41  13    7  17 

3  —    3  — 

—    3  29    3 

3  -    3  43 
17    3  —    3 
13  17     7  — 

7    3  11     3 

-19 

3  -    3  - 

77 
78 
79 

3  -    3  13 
29    3  37    3 
—    7  —  11 

11     3  -    3 
73  13  -     7 
3  41     3  - 

7 59 

3  -    3  - 
89    3  -    3 

3  11    3  71 

41    3  17    3 

7 17 

—  3  61     3 

—  11     7  47 
3  13    3  - 

80 

3  53    3  - 

-3-3 

13  71  23     7 

3  29    3  - 

11     3  13    3 

81 
82 
83 

-    3  11    3 
59  13  29  — 
3  19    3    7 

-  7  —  23 
3  43    3  - 

—  3—3 

3  -    3  11 
—    3  19    3 
53     7  11  — 

47    3  79    3 
3  13    3  31 

7  17  —  29 
3  —    3  73 
19    3  17    3 

84 
85 
86 

31    3     7    3 

-  11  47  67 
3     7    3- 

13  47  19  - 
3-37 
79    3     7    3 

3  -    3  - 
-3-3 
37 

—    3  11    3 

19    7 

3  89    3  53 

23 7 

3  -    3  83 
—    3-3 

87 
88 
89 

7    3-3 

13 23 

3  29    3  59 

31  —  23  — 
3     7    3  — 
7    3  37    3 

3  11     3     7 
—    373 
11  -  79  - 

—  3—3 

—  11 

3—37 

—  7  —  13 
3  37    3  - 

—  3  23    3 

90 

—    3-3 

71  29 

3     7    3  — 

11    3     7    3 

83  - 

91 
92 
93 

19—    7  - 
3  —    3  — 
71    3  41    3 

3  13    3  11 
61     3  13    3 
-  67     7  - 

7    3-3 

-  23  -  11 

3  -    3  19 

23 13 

3     7    3- 
7    3-3 

3  41     3     7 
—    373 
13  — 

94 
95 
96 

7  -  23  97 

3  13    3  37 

-    3  13    3 

3  -    3  - 

-    3  31     3 

7  —  59  — 

—  3  11     3 

—  89    7  13 
3  —    3  — 

3  -    3  - 
-    3  23    3 

3     7    3  11 

7    3-3 

31  -  11  - 

97 
98 
99 

89  31  17    7 

3  —    3  17 

-    3  -    3 

3  11     3  — 
-3-3 
11  23  47    7 

-    3  71     3 
7  11  31  - 
3  —    3  — 

37  -    7  - 

3  -    3  - 

—    3  19    3 

3  -    3  - 
13    3  43    3 
-  61     7  - 

XXIX.     LEAST  DIVISORS. 


85 


5 

6 

7 

8 

9 

n 

1 

3    7 

9 

13    7    9 

1 

3    7    9 

1    3 

7 

9 

1 

3    7    9 

50 

— 

31  13 

— 

3  61     3  37 

TT 

3  —    3 

-  13 

_ 

7 

3 

11    3  — 

51 

3 

—    3 

7 

13    3  —    3 



7  31  - 

3  71 

3 



29 

3  —    3 

52 

59 

3     7 

3 

—  19  23  11 

3 

—    3 

17 

3 

11 

67  —    7 

53 

— 

53  11  23 

3  31    3     7 

41 

3  19    3 

—    7 

— 

17 

3 

54 

3 

7    3 

53 

43    3     7    3 



13 

3  - 

3 

11 

17 

3  23     3 

55 

7 

3  — 

3 

67  —  19  — 

3 

-    3     7 

-    3  37 

3 



7  29  11 

56 

— 



— 

3     7    3  — 

53 

3     7    3 

13  — 

11 

3 

-    3  41 

57 

3 

11    3 

13 

7    3  73    3 

29  23  53  — 

3  — 

3 

7 

— 

3  11     3 

58 

— 

3  — 

3 

-  11 

3 

7    3  — 

—    3 

7 

3 

43 

71  -  17 

59 

11 

—    7  59 

3  67    3  47 

7 

3  43    3 

—  31 

— 

53 

3 

13    3     7 

60 

~3 

—    3 

73 

11    3-    3 

13 

—  59  — 

3     7 

3 





3     7    3 

61 



3  47 

3 

61  -     7  31 

3 

-    3  37 

7    3 

23 

3 

41 

11 

62 

7 

13  — 

11 

3  —    3  — 

3—3 

11  61 

19 

3 

7    3  — 

63 

3 

-    3 

— 

—    3  —    3 

23 

-    7  — 

3  13 

3 

7 

3-3 

64 

— 

3  11 

3 

7  23  29  — 

3 

—    3  11 

—    3 

13 

3 



43  73  67 

65 

— 

—  79 

7 

3  -    3  - 

— 

3—3 

—  29 

7 

11 

3 

19    3  — 

66 

3 

—    3 

— 

-    3  59     3 

7 

—  11  — 

3  41 

3 

— 

— 

3  37    3 

67 

43 

3  29 

3 

67    7 

3 

13    3  - 

-    3 

11 

3 



-    7  13 

68 

13 

7  — 

19 

3  —    3  — 

•s_ 

3  13    3 

7  — 

71 

83 

3  61     3—1 

69 

3 

17    3 

— 

—    3—3 

— 

19—7 

3  — 

3 

29 

— 

3-    3 

70 

TT 

3  — 

3 

23     7  37  — 

~T 

11     3  - 

73    3 

l9 

T 

7  41  47  31 1 

71 

23  17 

— 

3  13     3  67 

71 

3  —    3 

43  11 



7 

3 

—    3  23 

72 

3 

—    3 

7 

53    3  13    3 

11 

7  19  29 

3  — 

3 

37 

23 

3  -    3 

73 

— 

3     7 

3 

17  37  53  - 

3 

73     3  47 

11     3 

83 

3 

19 

-  13     7 

74 

— 

29  — 

— 

3  17    3     7 

31 

3  -     3 

-    7 

_ 



3 

59    3  - 

75 

3 

7    3 

— 

—    373 

67 

11 

3  — 

3 



3  71     3 

76 

7 

3  13 

3 

47  79  11  — 

3 

-    3     7 

-    3 

3 

— 

7  43  — 

77 

23 



— 

3     7    3  17 

19 

3     7    3 

31  43 

13 

— 

3 

-    3  11 

78 

3 

—     3 

29 

7    3—3 

17 



3  — 

3 

7 

13 

3  53    3 

79 

— 

3  73 

3 

19  -  31  13 

3 

7    3  79 

23    3 

7 

3 

61 

-  11  19 

80 

S3 

—    7 

— 

3  11     3  - 

7 

3  41     3 

—  59 





T 

—    37 

81 

3  31     3 

41 

-3-3 



11  13  - 

3     7 

3 

19 



3     7    3 

82 

37 

3  23 

3 

11  —     7  — 

3 

-    3  17 

7    3 

3 



43 

83 

7 

—  61 

13 

3  —    3  — 

11 

3—3 

17  83 

— 

3 

7    3  37 

84 

3 

79    3 

11 

-3-3 

43  37     7  61 

3  17 

3 

13 

7 

3  29    3 

85 

17 

3  43 

3 

7  -  13  11 

3 

—    3  23 

—    3  31 

3 

11 

13 

86 

41 

17  11 

7 

3  —    3  - 

13 

3  —    3 

—  19 

7 

— 

3 

—    3  — 

87 

3 

—    3 

19 

—    3  11    3 

7  31  67  - 

3  - 

3 

11 

59 

3  19    3 

88 

53 

3  17 

3 

7 

3 

19    3  13 

83    3 

3 

17 

—    7  11 

89 

— 

7  13 

17 

3  —    3  — 

3  47    3 

7  13 

11 

89 

3 

17    3  — 

90 

3 

11     3 

— 

13     3  -     3 

47  43  29     7 

3  31 

~3 

61 

— 

3  11    3 

91 

— 

3  - 

3 

—     7  89  53 

3 

—    3  67 

—    3 

— 

3 

7  29  17  - 1 

92 

11 

19  - 

47 

3  59     3  13 

73 

3—3 



37 

7 

3 

-    3  17 

93 

3 

47     3 

7 

11     3  17     3 

— 

7  —  83 

3  11 

3  41 

— 

3-3 

94 

13 

3     7 

3 

17 

3 

—    3  — 

19    3  53 

3 



11—7 

95 



41  19  11 

3  73     3     7 

17 

3  61     3 

11     7 

— 

43 

3 

53    3  29 

96 

3 

7    3 

13 

—    373 

19 

17 

3  23 

3 

— 

11 

3—3 

97 

7 

3  11 

3 

43  13 

3 

29    3     7 

—    3 

— 

3 

— 

7  97  41 

98 

— 

59  — 

— 

3     7    3  71 

— 

3     7     3 

41  — 

— 

11 

3 

13    3  19 

99 

3  37    3 

23 

7    3—3 

13 

-  11  17 

3  67 

3 

7 

97 

3  13    3 

86 


XXX.     EXPONENTIALS. 


n 

0 

1    2    8 

* 

5 

« 

7 

8    9 

.0 

1.000 

1.010  1.020  1.030 

1.041 

1.051 

1.062 

1.073 

1.083  1.094 

.1 
.2 
.3 

1.105 
1.221 
1.350 

1.116  1.127  1.139 
1.234  1.246  1.2.59 
1.363  1.377  1.391 

1.1.50 
1.271 
1.40^ 

1.162 
1.284 
1.419 

1.174 
1.297 
1.433 

1.185 
1.310 
1.448 

1.197  1.209 
1.323  1..336 
1.462  1.477 

.4 
.6 
.6 

1.492 
1.649 
1.S22 

1.507  1.522  1.537 
1.665  1.682  1.699 
1.840  1.859  1.878 

1.5.53 
1.716 
1.896 

1.568 
1.733 
1.916 

1.584 
1.751 
1.935 

1.600 
1.768 
1.954 

1.616  1.6.32 
1.786  1.804 
1.974  1.994 

.7 

.8 
.9 

2.014 
2.226 
2.460 

2.0.34  2.0.54  2075 
2.248  2.270  2.293 
2.484  2.509  2.53i 

2.096  2.117  2.138 
2.316  2.340  2.363 
2.560  2.586  2.612 

2.160  2.181  2.203 
2.387  2.411  2.435 
2.638  2.664  2.691 

1. 

2.71S 
7.389 
20.09 

3.004  3.320  3.669 
8.166  9.025  9.974 
22.20  24.53  27.11 

4.055  4.482  4.9.S3 
11.02  12.18  13.46 
29.96  33.12  36.60 

5.474  6.050  6.686 
14.88  16.44  18.17 
40.45  44.70  49.40 

4. 
5. 

6. 

54.60 
148.4 
403.4 

60.34  66.69  73.70 
164.0  181.3  200.3 
445.9  492.7  544.6 

81.45  90.02  99.48 
221.4  244.7  270.4 
601.8  665.1  735.1 

109.9  121.5  134.3 
298.9  330.3  365.0 
812.4  897.8  992.3 

7. 

8. 
9. 

1097 
2981 
8103 

1212  1,339  1480 
3294  3641  4024 
8955  9897  10938 

1636  1808  1998 
4447  4915  5432 
12088  1336014765 

2208  2441  2697 
6003  6634  7332 
16318  18034  19930 

1 

n 

0 

1   2   3 

4 

5 

6 

7 

8   9 

.0 

1.000 

*990  *980  *970 

*961 

*951 

*942 

*932 

*923  *914 

.1 

9 

.*3 

0.905 
0.819 
0.741 

896  887  878 
811  803  795 
733  726  719 

869 

787 
712 

861 
779 

70S 

852 
771 
698 

844 
763 
691 

835  827 
756  748 
684  677 

.4 
.5 
.6 

0.670 
0.607 
0.549 

664  657  651 
600  595  589 
543  538  533 

644 

583 
527 

638 
577 
522 

631 
571 
517 

625 
566 
512 

619  613 
560  554 
507  502 

.7 

.8 
.9 

0.497 
0.449 
0.407 

492  487  482 
445  440  436 
403  399  395 

477 
432 
391 

472 

427 
387 

468 
423 
383 

463 
419 
379 

458  454 
415  411 
375  372 

k 

3. 

0.368 
0.135 
0.0498 

333  301  273 
122  111   100 
450  408  369 

247 
*907 
334 

223 

*821 

302 

202 
♦743 
273 

183 

*672 
247 

165  150 

*608  *550 

224  202 

4. 
5. 
6. 

0.0183 

0.00674 

0.00248 

166  150  136 
610  552  499 
224  203  184 

123 

452 
166 

111 
409 
150 

101 
370 
136 

*910 
335 
123 

*823  *745 
303  274 
111  101 

7. 

8. 
9. 

0.000912 
0.000335 
0.000123 

825  747  676 
304  275  249 
112   101  091 

611 

225 
083 

5.53 
203 

075 

501 
184 
068 

453 
167 
061 

410  371 
151   136 
055  050 

n 

nc 

"J 

1 
e" 

1 
e   " 

e 

(IT 
2 

HIT 

e  3 

1 
2 
3 

2.718 
5.436 
8.154 

282   .367879 
564   .735759 
845  1.103638 

E.718 
1.649 
1.396 

.368 
.607 
.717 

4.81 
23.1 
111 

1 

4 
3 

.208 

.0432 

.00898 

4 
6 
6 

10.873 
13.591 
16.309 

127  1.471518 
409  1.839397 
691  2.207277 

1.284 
1.221 
1.181 

.779 
.819 
.847 

535 
257 
123 

5 
6 
92 

.00187 

.000388 

.0000807 

7 
8 
9 

19.027 
21.746 
24.464 

973  2.575156 
255  2.943036 
536  3.310915 

1.154 
1.133 
l.llS 

.867 

.883 
.895 

596 
286 
137 

10 

752 
9406 

.0000168 

.00000349 

.000000725 

EXPLANATION   OF  TABLES. 


Throughout  these  tables  the  figures  of  the  argument  are 
printed  in  thick  type,  the  initial  figures  being  printed  in  the 
left-hand  column,  and  the  terminal  figures  in  the  top  row. 
The  entry  is  found  in  the  intersection  of  the  row  of  the  initial 
figures  with  the  colvimn  of  the  terminal  figure.  A  bar  below 
a  terminal  5  or  0  shows  that  the  final  5  has  been  increased ; 
hence,  when  the  entry  is  further  contracted,  the  terminal  fig- 
ure ought  not  to  be  increased  by  one. 

I.  Common  Logarithms,  pp.  2-5. 

Pages  2  and  3  give  the  logarithm  to  fou'r  places  of  any 
sequence  of  three  significant  figures.  The  column  headed  d 
gives  the  difference  between  the  last  logarithm  of  the  row  and 
the  first  logarithm  of  the  next  row ;  it  facilitates  the  finding 
of  the  difference  between  any  two  successive  logarithms  in  the 
row.  The  small  table  at  the  bottom  of  page  3  gives  the  pro- 
portional parts  of  the  tabular  differences  from  4  to  23.  The 
tabular  difference  is  printed  in  the  top  row,  and  the  tenths  in 
the  left-hand  column. 

Pages  4  and  5  give  the  logarithms  of  any  sequence  of  4 
figures  from  1000  to  1900.  The  proportional  parts  of  the 
tabular  differences  are  given  in  the  right  column ;  only  the 
difference  is  printed,  the  tenth  being  understood  from  the  loca- 
tion of  the  proportional  part. 

The  small  table  at  the  bottom  of  page  5  gives  the  logarithm 
to  six  places  of  the  numbers  1.000  to  1.100  which  occur  in  cal- 
culations of  interest.  The  initial  pair  of  figures  are  given  only 
for  the  0  entry  and  are  understood  for  the  remaining  entries 
of  the  row ;  unless  an  asterisk  is  printed  in  front,  which  indi- 
cates that  the  initial  figures  are  those  printed  in  front  of  the 
next  row. 


88  EXPLANATION   OF   TABLES. 

Given  a  number  to  find  its  logarithm,  or,  to  use  the  table 
directly. 

The  characteristic  or  integral  part  of  the  required  logarithm 
is  obtained  by  counting  the  number  of  places  by  which  the 
first  figure  of  the  number  is  removed  to  the  left  or  to  the  right 
of  the  unit's  i)lace ;  if  to  the  left,  the  characteristic  is  posi- 
tive ;  if  to  the  right,  negative.  Thus  the  characteristic  of  1234 
is  3,  while  that  of  .01234  is  2. 

The  mantissa,  or  fractional  part  of  the  required  logarithm,  is 
obtained  from  the  table  thus.  If  the  number  has  not  more 
than  two  significant  figures,  then  the  mantissa  is  found  in  the 
column  headed  0.  If  there  are  three  significant  figures,  the 
mantissa  is  found  in  the  intersection  of  the  row  for  the  first 
two  figures  with  the  column  headed  by  the  third  figure.  For 
example,  the  mantissa  of  23  is  .3617,  that  of  234  is  .3G92. 

When  the  number  contains  four  significant  figures,  tlie  man- 
tissa can  be  obtained  directly  from  pages  4  and  5  provided  the 
number  is  not  greater  than  1900.  AVhen  the  number  is  greater, 
the  required  mantissa  is  found  from  pages  2  and  3  by  interpola- 
tion. Find  the  mantissa  for  the  first  three  significant  figures ; 
find  the  difference  between  it  and  the  next  higher  mantissa  in 
the  table  (it  will  in  general  be  nearly  equal  to  that  printed  under 
d)  ;  find  from  the  table  at  the  bottom  of  page  3  the  proportional 
part  of  this  tabular  difference  for  the  fourth  significant  figure 
and  add  it  to  the  lower  mantissa.  Thus  log  2345  is  .3692  plus 
five-tenths  of  19,  that  is  10 ;  hence  .3702.  For  log  23456  we 
add  besides  six-hundredths  of  19,  that  is  1 ;  hence  .3703.  Or 
we  multiply  19  by  .56  aud  take  the  nearest  integer  to  the  value, 
namely,  11. 

Given  a  logarithm  to  Jiyul  the  corresponding  number;  or,  to 
use  the  table  inversely. 

When  the  number  is  wanted  to  not  more  than  three  significant 
figures,  find  the  mantissa  in  the  table  which  is  nearest  to  the 
given  mantissa ;  the  corresponding  argument  gives  three  figures 
of  the  number,  and  the  position  of  the  decimal  point  is  deter- 
mined by  the  characteristic.  Thus  the  number  corresponding 
to  2.5015  is  317,  that  to  2.5020  is  .0318. 

Suppose  four  significant  figures  are  wanted.  If  the  mantissa 
does  not  exceed  .2785,  the  nearest  mantissa  on  pages  4-5  will 
point  out  the  number  to  four  figures.     If  it  exceeds  the  above 


EXPLANATION    OF   TABLES.  89 

number,  the  fourth  figure  is  obtained  by  interpolation.  Find 
the  next  lower  mantissa ;  find  the  difference  between  the  said 
mantissa  and  the  next  higher,  also  the  difference  between  said 
mantissa  and  the  given  mantissa,  and  find  from  the  table  of 
page  3  what  proportional  part  the  latter  difference  is  of  the 
former.  For  example,  the  next  lower  to  .7370  is  .7364,  hence 
the  first  part  of  the  number  is  545,  and  the  fourth  figure  is 
that  part  of  ten  which  6  is  of  8 :  namely  7  ;  hence  5.457. 

To  find  the  logarithm  of  a  product. 

Take  the  sum  of  the  logarithms  of  the  factors.     Thus, 

log  (123x4567)=  2.0899 

3.6590 

6 


6.7495 
To  find  the  logarithm  of  a  quotient. 

Subtract  the  logarithm  of  the  denominator  from  the  loga- 
rithm of  the  numerator.     Thus, 

log  1^^  =  2.0899 
3.6596 

2.4303 

To  find  the  logarithn  of  a  poiver. 

Multiply  the  logarithm  of  the  base  by  the  index  of  the 
power.     Thus, 

log      9872  ^  2  (2.9943)  =  5.9886 
log  (.987)- =  2(1.9943)  =1.9886 

To  find  the  logarithm  of  a  root. 

Divide  the  logarithm  of  the  base  by  the  index  of  the  root. 
Thus, 

log  V987  =  i (2.9943)  =  1.4972 
log  V:987  =  i  (1.9943)  =  1.9972 

II.  Antilogarithms,  pp.  6-7. 

This  is  a  table  of  the  fractional  powers  of  10  from  10 -^^  to 
10-^.  The  first  two  figures  of  the  fraction  or  mantissa  are 
given  in  the  left  column,  and  the  third  in  the  top  row. 


90  EXPLANATION   OF   TABLES. 

To  find  the  number  corresponding  to  a  logarithm. 

This  is  given  by  a  direct  use  of  the  table.  Find  the  entry 
for  the  first  three  figures  of  the  mantissa,  take  the  difference 
between  that  entry  and  the  next  higher,  and  from  the  column 
of  proportional  parts  find  the  part  which  requires  to  be  added 
to  the  entry  on  account  of  the  fourth  figure.  Insert  the  deci- 
mal point  in  the  place  indicated  by  the  characteristic  of  the 
logarithm.  For  example,  to  find  the  antilogarithm  of  2.9876. 
For  987  we  have  9705,  difference  is  22 ;  the  proportional  part 
of  22  for  6  is  13,  therefore  9705  +  13  =  9718,  and  inserting  the 
decimal  point,  971.8. 

The  direct  use  of  a  table  of  antilogarithms  serves  the  same 
purpose  as  the  inverse  use  of  a  table  of  logarithms. 

III.   Addition  Logarithms,  pp.  8-9. 

The  argument  is  log  n,  where  n  is  a  fraction  less  than  unity. 
Thus  the  characteristic  of  log  n  is  negative ;  it  is  not  printed 
so,  but  is  indicated  by  its  complement  to  10.  Thus  the  argu- 
ment 9.713  means  9.713  -  10,  or  1.713. 

Given  the  logarithm  of  each  of  two  numbers,  to  find  the  loga- 
rithm of  their  sum. 

Let  a  and  b  denote  the  two  numbers,  of  which  b  is  the  less. 
Then 

log  (a  +  6)  =  log  a  +  log^'l  +  ^\  (1) 

and  log  n  =  log  -  =  log  6  —  log  a.  (2) 

a 

Having  found  log  n  by  means  of  (2),  we  get  log  (1  +  n)  from 

the  table,  and  by  adding  it  to  log  a  obtain  log  (a  +6). 

Example,  log  a  =  3.8060,  log  &  =  2.1618. 

Therefore,  log  n  =  log-  =  2.1618 

"     3.8060 
2.3558 
therefore,  from  the  table,  log  (1  +  n)  =  0.0097 
therefore,  log  (a  +  6)  =  3.8060 

0.0097 


3.8157 


EXPLANATION   OF   TABLES.  91 

To  solve  the  same  question  by  means  of  Tables  I.  and  II. : 
a  =  6397,   b  =  1449  +  2  =  145.1, 
a  +  6  =  6542,   log  (a  +  6)  =  8156  + 1  =  3.8157. 

IV.  Subtraction  Logarithms,  pp.  10-13. 

This  table  is  arranged  similarly  to  the  preceding,  excepting 
that  the  tenth  decade  is  expanded  on  pages  12-13. 

Given  the  logarithm  of  each  of  two  numbers,  to  find  the  loga- 
rithm of  their  difference. 

Let  a  and  6  denote  the  two  numbers,  of  which  b  is  the  less. 
Then, 

log  (a  -  6)  =  log  a(l  _  ^^  =  log  a  +  log  (l  -^^ 

=  loga— log 

1-^ 
a 

Let  log  n  =  log  -,  then  from  the  table  we  get  log ,  which 

a  1  —  n 

subtracted  from  log  a,  gives  log  (a  —  6). 

Example.     Given  log  a  =  1.9876   and  log6  =  L5432. 

Then  log  n  =  log- =  1.5432 

"      1.9876 


1.5556  i.e.,  9.5556  -  10; 
Now,  from  the  table,  9.555  gives  0.1931  and  6  gives  3, 

...  log^—     =0.1934, 
1  —  n 

.-.  log(a-&)=L9876 
0.1934 

1.7942 

Suppose  that  log  (1  +  n)  is  known  (n  being  less  than  unity) 

and  its  position  in  Table  III.,  then  log can  be  found  in 

1  —  n 

the  corresponding  position  in  Table  IV.,  and  log  can  be 

found  by  adding  these  two  logarithms  together. 

Logarithms  of  addition  and  subtraction  are  sometimes  called 
Gaussian  logarithms. 


92  EXPLANATION   OF  TABLES. 


V.  Logarithmic  Sines  and  Cosines,  pp.  14-17.. 

Pages  14-15  give  the  logarithm  of  the  sine  to  every  tenth  of 
a  degree,  that  is,  every  six  minutes.  The  logarithm  of  the 
cosine  is  obtained  by  taking  the  riglit-hand  argument  and  read- 
ing backwards.  The  d  column  gives  the  difference  between 
the  two  last  entries  of  a  row,  the  last  entry  of  one  row  being 
identical  with  the  first  of  the  succeeding.  As  the  sines  and 
cosines  are  all  less  than  unity,  the  characteristics  of  the  loga- 
rithms are  all  negative ;  they  are  indicated  by  their  complement 
to  10.  Pages  16-17  give  the  sines  and  cosines  for  the  first 
nine  degrees  to  every  hundredth  of  a  degree. 

Given  an  arc,  to  find  its  log  sin. 

If  the  arc  is  less  than  90°,  its  log  sin  is  found  by  the  direct 
use  of  the  table.  Por  example,  to  find  log  sin  17°.66.  By  the 
table  log  sin  17°.6  is  9.4805-10,  the  difference  is  24,  and 
the  proportional  part  for  6  is  14 ;  hence,  9.4819  —  10.  If  the 
arc  is  >  90°  but  <  180°,  find  the  log  sin  of  the  difference  be- 
tween 180°  and  the  arc ;  if  >  180°  but  <  270°  find  that  of  the 
difference  between  the  arc  and  180°,  and  if  >  270°  but  <  360°, 
find  that  of  the  difference  between  360°  and  the  arc. 

Given  the  log  sin,  to  find  the  arc. 

Por  example,  to  find  the  arc  in  degrees  the  log  sin  of  which 
is  9.6669.  The  next  lower  log  sin  in  the  table  is  9.6659,  which 
corresponds  to  27°. 6 ;  the  tabular  difference  is  14,  and  the  given 
difference  is  10 ;  hence,  the  arc  is  27°. 67. 

At  the  bottom  of  page  17  there  are  two  auxiliary  tables.  The 
one  gives  the  equivalent  in  minutes  of  the  fractions  of  a  de- 
gree ;  thus,  0°.63  is  equivalent  to  37 '.8.  The  other  is  called  a 
Delambre's  table ;  it  is  used  to  find  the  log  sin  of  a  small  arc. 
On  account  of  the  table,  pages  16-17,  this  auxiliary  table  is  not 
required,  except  Avhen  the  arc  is  less  than  0°.4.  By  S  is  here 
meant  the  logarithm  of  the  ratio  of  the  number  expressing  the 
degree  to  the  number  expressing  the  corresponding  sine. 

To  find  the  sine  of  a  small  arc. 
Let  n  denote  the  number  of  degrees  ;  then, 
log  sin  n°  =  log  n  —  S. 


EXPLANATION   OF   TABLES..  93 

Example  :  to  find  log  sin  0°.123.  Log  .123  is  9.0899  —  10,  and 
S  is  1.7581 ;  hence,  logsin0°.123  is  7.3318  -  10. 

To  find  a  small  arc,  given  its  log  sin. 

We  have  log  n  =  log  sin  91°  +  S. 

For  example,  given  log  sin  to  be  7.1234  — 10.  As  the  log  sin 
is  less  than  8.2872-10,  the  value  of  S  to  add  is  1.7581. 
Hence,  8.8815  — 10,  the  number  corresponding  to  which  is 
.0761,  hence  0°.0761. 

VI.  Logarithmic  Tangents  and  Cotangents,  pp.  18-21. 

This  table  is  similar  to  the  preceding.  After  45°  the  tan- 
gent is  greater  than  unity,  the  characteristic  is  no  longer 
negative ;  hence,  the  true  characteristic  is  printed. 

Of  the  two  auxiliary  tables  at  the  bottom  of  page  21,  the  one 
gives  the  equivalent  in  degrees  of  so  many  minutes ;  the  other 
is  a  Delambre's  table  of  T  for  the  first  four  degrees.  By  T  is 
here  meant  the  logarithm  of  the  ratio  of  the  number  of  degrees 
to  the  number  expressing  the  tangent.  It  is  used  in  finding 
the  log  tan  of  a  very  small  arc.  We  have, 
log  tan  n°  =  log  n—  T 
and  log  n  =  log  tan  n°  +  T. 

VII.  Logarithmic  Sines  and  Cosines  for  Minutes,  pp. 

22-23. 

Here  the  log  sin  is  given  directly  to  every  ten  minutes,  and 
by  interpolation  to  every  minute.  The  same  table  gives  log  cos 
when  read  backwards.  Pages  22-25  give  the  proportional 
parts  for  all  the  differences  from  1  to  100. 

Example:  to  find  log  cos  19°  28'.  Log  cos  19°  20'  is  9.9748 
— 10,  the  tabular  difference  is  —5,  the  proportional  part  of  —5 
for  8  is  -4;  hence,  9.9744-10. 

At  the  end  we  have  a  table  of  S  for  the  range  between  0°  and 
7°,  where  the  change  in  the  value  of  the  tabular  difference 
is  too  rapid  to  allow  of  interpolation  by  proportional  parts. 
Here  S  is  the  logarithm  of  the  ratio  of  the  number  of  minutes 
expressing  the  arc  to  the  sine  of  the  arc.  Thus,  3.5372  is  the 
log  of  the  ratio  of  378  to  .1097. 


94  EXPLANATION    OF   TABLES. 

VIII.  Logarithmic  Tangents  and  Cotangents  for 
Minutes,  pp.  24-25. 

Tills  table  is  similar  to  the  preceding. 

By  T  is  meant  the  log  of  the  ratio  of  the  number  of  minutes 
expressing  the  arc  to  the  tangent  of  the  arc. 

IX.  Natural  Sines  and  Cosines,  pp.  26-27. 

The  natural  sine  is  given  to  each  tenth  of  a  degree ;  that 
is,  to  every  six  minutes.  The  equivalent  minutes  are  printed 
alongside  of  the  tenths  of  a  degree.  At  the  bottom  of  page  27 
there  is  a  table  of  proportional  parts,  the  whole  interval  being 
six,  to  facilitate  the  interpolation  to  a  minute. 

What  is  the  sine  of  34°  46'  ?  The  sine  of  34°  42'  is  .5693, 
the  tabular  difference  is  14,  and  the  pp.  of  14  for  4  is  9 ;  hence, 
.5702. 

What  is  the  arc  whose  cosine  is  .4326  ?  The  arc  of  .4321  is 
64°  24',  the  tabular  difference  is  16,  the  difference  of  given 
cosine  is  5,  corresponding  to  a  pp.  of  5  for  a  tabular  difference 
of  16  we  have  2' ;  hence,  the  arc  is  64°  22'. 

X.  Natural  Tangents  and  Cotangents,  pp.  28-29. 
XI.  Natural  Secants  and  Cosecants,  pp.  30-31. 

These  tables  are  similar  to  the  preceding.  At  the  end  of 
each  we  have  a  continuation  of  the  table  of  proportional  parts, 
the  interval  being  six. 

XII.  Radians,  pp.  32-33. 

By  a  radian  is  meant  the  unit  of  circular  measure  of  an 
angle.  The  table  gives  directly  the  number  of  radians  equiv- 
alent to  any  number  of  degrees  expressed  by  not  more  than 
three  significant  figures.  The  integer  figure  of  the  entry  is 
printed  only  in  the  0  column.  Thus,  the  equivalent  of  67°.8  is 
1.1833  radians,  and  the  equivalent  of  1  radian  is  57°.3.  The 
tabular  difference  is  either  17  or  18 ;  hence,  to  find  the  equiva- 
lent for  4  significant  figures,  we  add  the  proper  pp.  of  either  18 
or  17,  as  the  case  may  be. 

The  column  headed  h  m  gives  the  equivalent  in  hours  and 
minutes  of  the  corresponding  number  of  degrees  in  the  left 


EXPLANATION    OF   TABLES.  95 

column ;  and  the  adjacent  column  headed  p  gives  that  fraction 
of  a  whole  period  or  perigon  which  is  equivalent  to  the  ratio  of 
the  corresponding  number  of  degrees  to  360°. 

The  small  table  at  bottom  of  page  33  gives  the  number  of 
radians  equivalent  to  the  given  number  of  minutes,  while  the 
column  headed  p  gives  that  fraction  of  a  period  or  perigon 
which  the  corresponding  number  of  minutes  bears  to  360°. 

When  the  decimal  point  is  changed  by  any  number  of  places 
in  the  argument,  the  decimal  point  is  changed  by  an  equal 
number  of  places  in  the  entry.     Thus, 

3°.6=. 06283  and  360°=  6.283. 

XIII.   Reciprocals,  pp.  34-35. 

The  reciprocal  is  given  directly  for  any  sequence  of  three 
figures,  the  decimal  point  being  after  the  first.  When  the 
decimal  point  in  the  argument  is  shifted  any  number  of  places, 
the  decimal  point  in  the  entry  is  shifted  an  equal  number  of 
places  in  the  opposite  direction.     Thus, 

J_  = .  1267,  ^  =  .01267, 

7.89  '  78.9 

-^  =  .001267,  —  =  1.267. 

789  '  .789 

At  the  bottom  of  page  35  we  have  the  first  nine  multiples  of 
the  fractions  \,  \,  etc.,  up  to  ^.  A  bracket  indicates  that  the 
figures  included  repeat  themselves. 

XIV.   Squares,  pp.  36-37. 

This  table  gives  directly  to  four  significant  figures  the  square 
of  any  sequence  of  three  figures,  the  decimal  point  being  after 
the  first.  When  the  decimal  point  changes  in  the  number,  the 
decimal  point  in  the  square  changes  by  double  the  number  of 
places  in  the  same  direction.  Thus  the  square  of  3.76  is  14.14, 
that  of  37.6  is  1414,  and  that  of  .376  is  .1414. 

When  the  number  consists  of  more  than  three  figures,  the 
square  may  be  found  by  means  of  the  table  of  proportional 
parts.  For  example,  to  find  the  square  of  1889  to  four  signi- 
ficant figures.  The  square  of  188  is  35,340,  the  pp.  of  38  for  9  is 
34 ;  therefore  the  square  of  1889  is  3,568,000.  Here  the  zeros 
are  not  significant,  but  only  indicate  the  position  of  the  decimal 
point. 


96  EXPLANATION   OF   TABLES. 

To  find  the  complete  square  for  any  sequence  of  three  figures. 

The  complete  square  of  any  two  figures  is  given  in  the  zero 
eohimn.  If  the  number  of  three  figures  is  less  than  317,  we 
have  to  find  the  square  of  the  third  figure,  and  a})pend  the 
terminal  figui'e  to  the  entry  of  the  table,  diminishing  the  ter- 
minal figure  of  the  entry  by  one  if  the  number  appended  is 
equal  to  or  greater  than  5.  For  instance,  take  234.  The 
square  of  4  is  16,  hence  6  is  to  be  appended  to  5476,  but  the 
fourth  figure  reduces  to  5  because  it  has  been  increased  by 
one  when  the  6  was  cut  off.  Hence  the  complete  square  is 
54,756.  When  the  number  exceeds  316,  find  the  square  of  the 
two  terminal  figures  in  the  zero  column,  take  the  last  two 
figures  of  it  and  append  them  to  the  entry,  diminishing  the 
terminal  figure  of  the  entry  by  one  if  the  addendum  equals  or 
exceeds  50.  For  example,  the  last  two  figures  of  the  square 
of  96  is  16,  the  entry  for  896  is  8028,  hence  the  complete 
square  is  802,816. 

The  table  at  the  bottom  of  page  37  gives  the  square  of  the 
reciprocal  of  any  number   of  two   digits.     Thus   the   square 

of  —  is  .0865.      When  the  decimal  point  is  shifted  in  the 

3.4  ^ 

argument,  the  decimal  point  of  the  entry  requires  to  be  shifted 
by  twice  the  number  of  places  in  the  opposite  direction.     Thus 

the  square  of  -—  is  1.32. 

.87 


XV.  Cubes,  pp.  38-39. 

This  table  gives  to  four  figures  the  cube  of  any  number  of 
three  figures,  and  in  the  0  column  the  complete  cube  of  any 
number  of  two  figures.  When  the  decimal  point  is  shifted  in 
the  number,  the  decimal  point  of  the  cube  requires  to  be  shifted 
thrice  the  number  of  places  in  the  same  direction.  Thus  the 
cube  of  1.23  is  1.861,  that  of  12.3  is  1801,  that  of  .123  is 
.001861. 

The  small  table  of  page  39  gives  the  cube  of  the  reciprocal  of 

the   number.     Thus  the   cube  of  —  is  .00142.      When   the 

8.9 
decimal  point  is  shifted  in  the  number,  the  decimal  point  in 
the  reciprocal  of  the  cube  is  shifted  thrice  the  number  of 
places  in  the  opposite  direction. 


I 


EXPLANATION   OF   TABLES.  97 


XVI.  Square  Roots,  pp.  40-43. 

Tlie  first  part  of  the  table,  pages  40-41,  gives  the  square  root 
of  any  number  of  three  significant  figures,  when  the  decimal 
point  is  after  the  first  figure,  or  is  any  even  number  of  places 
to  the  right  or  left  of  that  position ;  while  the  second  part  of 
the  table,  pages  42-43,  gives  the  square  root,  when  the  decimal 
point  is  after  the  second  figure  or  any  even  number  of  places  to 
the  right  or  left  of  that  position.  The  square  root  is  given  to 
five  figures,  the  initial  figure  being  printed  only  in  the  0  column. 

When  the  decimal  point  of  the  number  is  shifted  any  even 
number  of  j)laces  from  its  position  after  either  the  first  digit 
or  after  the  second  digit,  the  decimal  point  in  the  correspond- 
ing entry  shifts  by  half  the  number  of  places  in  the  same 
direction.     Thus, 

Va87=  3.1417,  V98J  =  9.9348,      V9S7  =  31.417, 
V9870  =  99.348,  Vi987  =  .99348,  V.0987  =  .31417. 
The  small  table  of  page  41  gives  the  square  root  of  the  recipro- 
cal of  any  number  of  two  figures,  the  decimal  point  being  after 
the  first  figure ;  while  the  small  table  of  page  43  gives  the  same 
when  the  decimal  point  is  after  the  second  figure.     Thus, 

_4:_  =  .319     Ji_  =  .101,      J--  =  1.01,    —4=  =  .0319. 

V9.8  V98  V.98  V980 

XVII.  Cube  Roots,  pp.  44-49. 

The  first  part  of  the  table  gives  the  cube  root,  when  the 
decimal  point  is  after  the  first  significant  figure,  or  when  dis- 
placed any  multiple  of  three  places  to  the  right  or  left  of  that 
position ;  the  second  part  similarly  when  the  decimal  point  is 
after  the  second  significant  figure  ;  and  the  third  part  when  it 
is  after  the  third.  A  displacement  of  three  places  in  the  num- 
ber causes  a  displacement  of  one  place  in  the  same  direction 
in  the  cube  root.     Thus, 

a/i:23  =  1.0714,     ■v/12:3  =  2.3084,     a/123  =  4.9732, 
■v/i230  =  10.714,    -v/:0l23  =  .23084,    -v^m  =  .49732. 

Similarly  the  three  small  tables  give  the  cube  root  of  the 
reciprocal  of  any  two  figures  for  the  three  distinct  positions 
of  the  decimal  point. 


98  EXPLANATION    OF   TABLES. 

XVIII.   Multiples,  pp.  50-67. 

This  table  gives  tlie  first  nine  multiples  of  any  number  of 
three  figures,  and  the  folding  table  at  the  end  gives  the  same 
for  any  number  of  two  figures.  By  means  of  this  table  and 
our  knowledge  of  the  ordinary  multiplication  table  we  can 
write  down  any  of  the  nine  multiples  of  a  number  of  four 
figures,  and  with  the  help  of  the  folding  table  we  can  do  the 
same  for  any  number  of  five  figures.  By  a  double  reference  to 
the  table  we  obtain  a  multiple  of  six  figures,  and  so  on.     Thus, 

8  times        789  =     6312 

8  times      6789  =       6312 
48 


54312 


8  times    56789=       6312 
448 
454332 

8  times  456789  =  6312 

3648 
3654312 

To  multiply  any  tivo  numbers  together. 

Consider,  for  example,  the  product  of  123,456,789  and  6987. 
Turn  up  the  multiples  of  789,  and  write  down  the  7,  8,  9,  and  6 
multiples  under  one  another  in  the  usual  manner,  only  space  is 
to  be  left  between  each  pair  of  multiples  for  another  row  of 
figures ;  then  turn  up  the  multiples  of  456,  write  down  the  7 
multiple  with  its  initial  figure  below  the  fourth  figure  of  the 
7  multiple  of  789,  and  similarly  for  the  other  multiples ;  then 
turn  up  the  multiples  of  123,  write  down  the  7  multiple  with 
its  first  figure  above  the  fourth  figure  of  the  7  multiple  of  456, 
and  so  on,  as  follows  .• 

861    5523 
3192 
984    6312 
3648 
1107    7101 

4104 
738    4734 

2736 

86259258474  3 


EXPLANATION   OF   TABLES.  99 

To  divide  one  number  by  another. 

For  example,  to  divide  4,567,890  by  567.  Turn  up  the  mul- 
tiples of  567  ;  find  the  next  lower  to  4567,  deduct  it ;  take  down 
another  figure,  find  the  next  lower  multiple  to  the  number  so 
formed,  and  so  on,  as  follows  : 

567)4567890(8056 
4536 


If  the  divisor  consist  of  four  figures,  as  5678,  turn  up  the 
multiples  of  567  and  correct  them  mentally  for  the  additional 
figure  8.  If  there  are  five  figures,  as  56,789,  correct  the  multi- 
ples of  567  by  adding  the  multiples  of  89  from  the  folding  table. 

XIX.  Circumference  of  Circle,  pp.  68-69. 

When  the  decimal  point  is  changed  in  the  diameter,  the 
decimal  point  in  the  circumference  changes  by  an  equal  num- 
ber of  places  in  the  same  direction.  When  n  represents  the 
radius,  the  circumference  is  obtained  by  doubling  the  entry. 

The  small  table,  page  69,  gives  the  value,  the  logarithm,  and 
the  reciprocal  of  frequently  occurring  constants,  which  involve 
TT.  The  mantissa  of  the  logarithm  of  the  reciprocal  is  the  com- 
plement to  1  of  the  mantissa  of  the  logarithm  of  the  constant. 

Thus,  log  -  is  1.5029. 

XX.  Area  of  Circle,  pp.  70-71. 

When  n  denotes  the  radius,  the  area  is  obtained  by  multi- 
plying the  entry  by  4.  When  the  decimal  point  is  changed  in 
the  diameter,  the  decimal  point  of  the  area  changes  by  double 
the  number  of  places  in  the  same  direction.  Thus,  when  the 
diameter  is  3.96  the  area  is  12.32,  when  39.6  then  1232,  when 
.396  then  .1232. 

The  diameter  of  a  circle  of  given  area  is  obtained  by  the 
inverse  use  of  the  table. 

When  n  denotes  the  diameter  of  a  sphere,  the  surface  is 


100  EXPLANATION   OF   TABLES. 

ttw*.     Hence  the  surface  of  a  sphere  of  given  diameter  is  ob- 
tained by  multiplying  the  entry  of  the  table  by  4. 

The  auxiliary  table,  page  71,  gives  the  decimal  equivalents 
of  the  binary  divisions  of  the  inch,  and  also  the  decimal  equiv- 
alents of  a  number  of  inches  as  part  of  the  foot  or  of  the 
yard.  Thus,  the  area  of  a  circle  of  3f  inch  diameter,  is  that 
of  3.375  inch;  hence,  8.920  +  26,  that  is,  8.946  square  inches. 

XXI.  Content  of  Sphere,  pp.  72-73. 

This  table  gives  the  content  of  a  sphere  of  which  n  is  the 
diameter.  When  the  radius  is  given,  the  spherical  content  is 
obtained  by  multiplying  the  tabular  entry  by  8.  When  the 
decimal  point  is  changed  in  the  diameter,  the  decimal  point  of 
the  content  is  changed  thrice  the  number  of  places  in  the  same 
direction. 

The  small  table  at  the  bottom  of  page  73  gives  the  logarithm 
of  the  product  of  successive  integers  from  1  up  to  n,  and  the 
logarithm  of  the  powers  of  2  up  to  the  29th.  For  example, 
logl.2.3.4.5.6  is  2.8573. 

XXII.  Hyperbolic  Logarithms,  pp.  74-75. 

This  table  gives  directly  the  hyperbolic  or  natural  logarithm 
of  any  sequence  of  three  signiticant  figures,  the  decimal  point 
being  after  the  first.  When  the  decimal  point  of  the  sequence 
is  shifted  n  places  to  the  right  from  the  above  position,  find 
the  logarithm  of  10"  in  the  auxiliary  table  and  add  it  to  the 
entry ;  and  when  the  decimal  point  is  shifted  n  places  to  the 
left,  add  the  logarithm  of  10"".     Thus, 

log  56.7  =  1.7352  log  567  =  1.7352 

2.3026  4.6052 


4.0378  6.3404 

log  .567  =  1.7352  log.  .0507  =  1.7352 

3.6974  5.3948 


1.4326  3.1300 

By  m  in  the  auxiliary  table  is  meant  the  modulus  or  multi- 
plier for  converting  the  natural  or  hyperbolic  logarithm  of  a 

number  into  the  common  logarithm  of  the  number,  and  by  —  is 

m 
meant  the  reciprocal  modulus  or  multiplier  for  converting  the 
common  log  of  a  number  into  the  natural. 


EXPLANATION   OF   TABLES.  101 

Example :  To  fiud  the  hyperbolic  log  of  1889,  given  the  com- 
mon log  to  be  3.2762. 

The  equivalent  of  3  =6.9078 
The  equivalent  of  .2  =  0.4605 
The  equivalent  of  .07  =  0.1612 
The  equivalent  of  .006  =  0.0138 
The  equivalent  of  .0002  =  0.0005 

Therefore  hyp.  log.  of  1889  =  7.5438 

XXIII.  Amount  of  One  Unit  of  Money  at  the  End  of 

a  Given  Number  of  Years,  p.  76. 

The  argument  in  the  left-hand  column  is  the  number  of 

years  during  which  one  unit  of  money  (whether  dollar,  pound, 

franc,  or  mark)  has  been  allowed  to  accumulate  at  compound 

interest,  while  the  argument  in  the  top  row  is  the  rate  of 

interest  expressed  as  so  much  per  cent  per  year.     The  general 

expression  for  the  amount  of  one  unit  in  n  years  at  r  per  cent 

per  year  is  fl  +  ^^ 

• 
Given  the  prindixd,  the  number  of  years,  and  the  rate,  to  find 

the  amount. 

Find  from  the  table  the  amount  of  one  unit  of  money  for  the 

given  number  of  years  and  rate,  and  multiply  that  number  by 

the  principal.     For  example,  to  find  the  amount  of  $123  at  the 

end  of  25  years  at  6  per  cent  per  year.     The  entry  for  25  years 

and  6  per  cent  is  4.292 ;  to  find  the  i)roduct  of  this  number 

and  123,  turn  up  the  multiples  of  123. 

4.292  X  123  =        246 
1107 
24  6 
492 
527.916 

As  the  fourth  figure  of  4.292  is  inexact,  the  figures  1  and  6 
of  the  product  are  not  significant;  hence  the  result  is  $527.9. 

Given  the  amount,  the  number  of  years,  and  the  rate,  to  find 
the  principal. 

Find  from  the  table  the  amount  of  one  unit  for  the  given 
number  of  years  and  rate,  and  divide  the  total  amount  by  it ; 
the  quotient  is  the  principal. 


102  EXPLANATION    OF   TABLES. 

Given  the  principal^  the  rate,  and  the  amount,  to  find  the  num- 
ber of  years. 

Divide  the  amount  by  the  principal  and  compare  the  quo- 
tient with  the  entries  in  the  column  under  the  given  rate.  For 
example,  to  find  the  number  of  years  in  which  $456  becomes 
1^742.82  at  5  per  cent  per  year.  Dividing  742.82  by  456,  we 
get  1.629,  which  is  the  entry  in  the  5  per  cent  column  for  10 
years. 

Given  the  principal,  the  amount,  and  the  member  of  years,  to 
find  the  rate. 

Divide  the  amount  by  the  principal  and  compare  the  quotient 
with  the  entries  in  the  row  of  the  given  number  of  years. 

To  find  the  amount  of  a  unit  of  money  for  a  number  of  years 
and  a  fraction  of  a  year. 

Find  the  difference  between  the  entry  for  the  number  of 
years  and  the  next  higher  entry,  multiply  it  by  the  fraction 
of  the  year,  and  add  the  resvdt  to  the  lower  entry.  For  ex- 
ample, to  find  it  for  7  years  and  3  months,  the  rate  of  interest 
being  8  per  cent.  The  entry  for  7  years  is  1.714,  and  that  for 
8  is  1.851 ;  the  difference  is  137,  the  fourth  part  of  which  is  34, 
which  added  to  1.714  gives  1.748. 

To  find  the  amount  of  a  unit  of  money  for  an  intermediate  rate 
of  interest. 

The  value  may  be  found  approximately  by  applying  the 
principle  of  proportional  parts  as  above.  For  example,  the 
amount  for  9  years  at  5^  per  cent  is  1.551,  plus  one-half  of 
138;  hence,  1.620. 

To  find  the  amount  of  one  unit  of  money  for  a  number  of 
years  greater  than  49. 

Break  the  number  of  years  into  parts  each  not  greater  than 
49,  multiply  together  the  entries  for  the  several  parts ;  the 
result  is  the  amount  for  the  given  number  of  years. 

For  example,  to  find  the  amount  of  one  unit  of  money  for  70 
years  at  10  per  cent.  The  entry  for  40  years  is  45.26,  and  that 
for  30  years  is  17.45.  To  find  the  product  of  45.26  by  17.45, 
turn  up  the  multiples  of  745  and  correct  them  for  the  1 ;  the 
result  is  789.7870,  but  the  last  three  figures  are  not  significant; 
hence,  789.8.     The  true  value  is  789.747. 


EXPLANATION   OF   TABLES.  103 

This  problem  may  also  be  solved  by  means  of  the  small 
table  of  logarithms,  page  5,  where  the  logs  of  the  coefficients 
from  1.000  up  to  1.100  are  given  to  six  places  in  order  that 
their  multiples  may  be  obtained  exact  to  four  places.  The  log 
of  1.10  is  .041393,  which  multiplied  by  70  gives  2.8975,  the 
antilogarithm  of  which  is  789.8. 


XXIV.  Present  Value  of  1000  Units  of  Money,  p.  77. 

This  table  gives  the  present  value  of  1000  units  of  money 
due  n  years  hence,  the  rate  of  interest  having  any  one  of  the 
values  in  the  top  row.  The  entry  is  given,  not  for  1,  but  for 
1000,  in  order  to  simplify  the  specification  of  the  decimal 
point.  When  an  entry  is  taken  out,  the  decimal  point  ought 
to  be  shifted  three  places  to  the  left. 

The  method  of  using  this  table  is  the  same  as  that  fop 
Table  XXIII. 

XXV.  Amount  of  an  Annuity  when  paid  at  the  End  of 
Bach  Year,  p.  78. 

This  table  gives  the  amount  of  an  annuity  of  one  unit  of 
money  per  year,  when  the  annuity  is  allowed  to  accumulate 
for  n  years,  the  first  payment  being  made  at  the  end  of  one 
year  from  the  time  of  reckoning. 

The  method  of  using  Table  XXIII,  applies  to  this  table,  "  one 
unit  of  money  per  year"  being  substituted  for  "one  unit  of 
money,"  excepting  the  rule  at  the  end  for  extending  the  table. 

In  order  to  extend  the  table,  the  value  of  ( 1  -| — —  ]   must  be 

found  by  that  rule,  and  the  result  substituted  in  the  formula 
at  the  top  of  the  table. 

XXVI.  Present  Value  of  the  Preceding,  p.  79. 

This  table  gives  the  value  at  the  beginning  of  the  time  of 
reckoning  of  an  annuity  of  one  unit  of  money  per  year  allowed 
to  accumulate  for  a  given  number  of  years,  the  first  payment 
being  made  at  the  end  of  one  year  from  the  beginning  of  the 
time  of  reckoning. 

The  method  of  using  the  table  is  the  same  as  for  Table  XXV. 


104  EXPLANATION  OF   TABLES. 


XXVII.  Amount  of  an  Annuity  when  paid  at  the  Be- 

ginning of  Each  Year,  p.  80. 

Tliis  table  gives  the  timount  of  au  annuity  of  one  unit  of 
money  per  year,  when  the  several  payments  are  allowed  to 
grow  at  any  one  of  the  rates  of  interest  specified,  the  first  pay- 
ment being  made  at  the  beginning  of  the  time  of  reckoning. 

The  method  of  using  the  table  is  the  same  as  for  Table  XXV. 

XXVIII.  Annuity   required   to    extinguish    a    Debt   of 

1000,  p.  81. 

This  table  gives  the  annual  sum  to  be  paid  for  a  given  num- 
ber of  years,  the  first  payment  being  made  one  year  from  the 
present  time,  in  order  to  extinguish  a  present  debt  of  1000 
units  of  money.  Here  the  1000  is  introduced  for  the  same 
reason  as  in  the  case  of  Table  XXIV. 

To  extend  the  table,  the  extended  value  of  (1  +  r)"  must  be 
found  and  substituted  in  the  formula  printed  at  the  top. 

XXIX.  Least  Divisors,  pp.  82-85. 

This  table  gives  the  least  divisor  of  any  number  up  to  10,000. 
The  first  two  figures  of  the  number  are  given  in  the  left-hand 
column,  the  third  figure  in  the  top  row  and  the  terminal  figure 
in  the  row  beneath.  The  only  terminal  figures  entered  are 
1,  3,  7,  9,  because  any  number  which  terminates  otherwise  is 
evidently  divisible  by  2  or  5. 

To  find  the  factors  of  any  number  less  than  10,000. 

If  it  is  an  even  number,  divide  out  2  until  the  remainder  is 
odd ;  if  it  then  ends  in  5,  divide  out  the  power  of  5 ;  then  enter 
the  table  with  the  remaining  quotient  to  find  its  least  divisor  ; 
divide  out  that  divisor,  and  with  the  then  remaining  quotient 
enter  the  table  again ;  and  so  on  until  the  remaining  quotient 
is  a  prime,  which  is  indicated  in  the  table  by  a  bar. 

Example  :  1889  is  a  prime. 

9876  =  2  X  4938  =  2^  x  2469. 

Xow  2469  has  least  divisor  3,  and  quotient  is  823,  and  823  is 
a  prime.     Hence, 

9876  =  22  X  3  X  823. 


EXPLANATION   OF   TABLES.  105 

XXX.   Exponentials,  p.  86. 

The  upper  part  of  the  table  contains  the  ascending  powers 
of  e  from  .00  to  .99  and  from  1.0  to  9.9;  and  the  lower  part 
the  corresponding  descending  powers.  The  upper  part  forms 
a  small  table  of  hyperbolic  antilogarithms. 

At  the  bottom  of  the  page  we  have  the  first  nine  multiples 
of  e  and  of  the  reciprocal  of  e,  the  first  nine  fractional  powers  of 
e,  both  positive  and  negative,  and  the  poAvers  of  e  given  by  the 

first  nine  multiples  of  ^,  both  positive  and  negative. 

XXXI.  Multiples  (Folding  Leaf). 

This  table  contains  the  first  nine  multiples  of  the  numbers 
from  1  to  99.  It  may  be  used  as  a  table  of  proportional  parts 
for  tenths  by  inserting  a  decimal  point  before  the  last  figure, 
and  for  hundredths  by  inserting  the  point  before  the  second 
last  figure. 


XXXI.     MULTIPLES. 


1 

0 

1   2   8 

4  5  6 

7   8  9 

10 

11  12  13 

14  15  16 

17  18  19 

0 

1   2   3 

4   5   6 

7   8   9 

•10 

11  12  13 

14"  15  16 

n   IS   19 

2 

0 

2   4   6 

8  10  12 

14  16  18 

20 

22  24  26 

28  30  32 

34  36  38 

3 

0 

3   6   9 

12  15  18 

21  24  27 

30 

33  36  39 

42  45  48 

51  54  57 

1 

0 

4   8  12 

16  20  24 

28  32  36 

40 

44  48  52 

56  60  64 

58  72  76 

5 

0 

5  10  15 

20  25  30 

35  40  45 

SO 

55  .»  65 

70  75  80 

85  90  95 

6 

0 

6  12  18 

24  30  36 

42  48  54 

60 

66  72  78 

84  90  96 

102  108  114 

7 

0 

7  14  21 

28  35  42 

49  56  63 

70 

77  34  91 

98  105  112 

119  126  133 

0 

8  16  24 

32  40  48 

56  64  72 

80 

88  %  1(M 

112  120  128 

136  144  152 

9 

0 

9  18  27 

36  45  54 

63  72  81 

90 

99  108  117 

126  135  144 

153  162  171 

20 

21  22  23 

24  25  26 

27  28  29 

80 

81  a   33 

34  35  36 

37  38  39 

1 

20 

21  22  23 

24  25  26 

27  28  29 

30 

31  32  33 

34  35  36 

37  38  39 

2 

40 

42  44  46 

48  SO  52 

54  56  58 

60 

62  .i4  66 

68  70  72 

74  76  78 

3 

60 

63  66  69 

72  75  78 

81  84  87 

90 

93  %  99 

102  105  108 

111  114  117 

4 

80 

84  88  92 

96  100  104 

108  112  116 

120 

124  128  132 

136  140  144 

148  152  156 

5 

100 

105  110  115 

120  125  130 

135  140  145 

ISO 

155  160  165 

170  175  180 

185  190  195 

6 

120 

126  132  138 

144  1.50  156 

162  168  174 

180 

186  192  198 

204  210  216 

222  228  234 

7 

140 

147  154  161 

163  175  182 

189  196  203 

210 

217  224  231 

238  245  252 

259  256  273 

8 

160 

168  176  184 

192  200  208 

216  224  232 

240 

248  256  264 

272  280  288 

296  304  312 

9 

ISO 

189  198  207 

216  225  234 

243  252  261 

270 

1279  2SS  297' 

306  315  324 

333  342  351 

40 

41  42  43 

44  45  46 

47  48  49 

50 

51  5i    53 

54  55  56 

57  58  59 

^ 

40 

41  42  43 

44  45  46 

47  48  49 

~5'(r 

51  52  53 

54  55  55 

57  58  59 

2 

82  84  86 

88  90  92 

94  96  98 

100 

102  lot  106 

108  110  112 

114  116  118 

3 

120 

123  126  129 

132  135  138 

141  144  147 

150 

153  156  159 

162  165  158 

171  174  177 

4 

160 

164  168  172 

176  180  184 

188  192  196 

200 

204  208  212 

216  220  224 

228  232  236 

5 

200 

205  210  215 

220  225  230 

235  240  245 

2.50 

255  260  265 

270  275  280 

285  290  295 

6 

240 

246  252  258 

264  270  276 

282  288  294 

300 

305  312  318 

324  330  336 

342  348  354 

7 

280 

287  294  301 

308  315  322 

329  336  343 

3.50 

357  364  371 

378  385  392 

399  406  413 

H 

320 

32S  336  344 

352  iM  3r,8 

376  384  392 

400 

408  416  424 

432  440  448 

456  464  472 

9 

360 

369  378  387 

396  405  414 

423  432  441 

450 

4.59  468  477 

486  495  S(M 

513  522  531 

(10 

61  62  63 

64  65  66 

67  68  69 

70 

71  73  78 

74  75  76 

77  78  79 

^ 

60 

61  62  63 

64  65  66 

67  68  69 

70 

71  72  73 

74  75  76 

77  78  79 

2 

120 

122  124  126 

128  1.30  132 

134  136  138 

140 

142  144  146 

148  150  152 

154  1.56  158 

183  186  189 

192  195  198 

201  204  207 

210 

213  216  219 

222  225  228 

231  234  237 

4 

240 

2U   248  252 

256  260  2f>4 

268  272  276 

280 

284  2SS  292 

296  300  304 

308  312  316 

9 

300 

.305  310  315 

320  325  3.TO 

335  340  345 

350 

355  iM  365 

370  375  380 

385  390  395 

6 

360 

366  372  378 

384  390  396 

402  408  414 

420 

426  432  438 

444  4.50  456 

462  468  474 

7 

420 

427  434  441 

448  455  462 

469  476  483 

490 

497  .504  511 

518  525  .532 

539  545  553 

488  496  504 

512  520  528 

536  .544  552 

560 

.568  576  584 

592  500  608 

616  624  632 

9 

540 

.549  558  567 

576  585  594 

603  612  621 

630 

639>  648  657 

565  675  584 

693  702  711 

80 

81  82  83 

84  85  86 

87  88  8!) 

90 

91  92  98 

94  95  96 

97  98  99 

1 

80 

81  82  83 

84  85  86 

87  88  89 

90 

,  91  92  93 

94  95  96 

97  98  99 

2 

160 

162  164  166 

168  170  172 

174  176  178 

ISO 

1S2  1S4  186 

188  190  192 

194  196  198 

3 

240 

243  246  249 

252  255  258 

261  264  267 

270 

273  276  279 

282  2S5  288 

291  2'H   297 

4 

320 

324  328  332 

336  MO  344 

348  352  3.S6 

360 

364  368  372 

375  380  384 

388  392  396 

5 

400 

405  410  415 

420  425  430 

.  435  -l-IO  445 

4.50 

455  460  465 

470  475  480 

485  4'X)  495 

» 

480 

486  492  498 

504  510  516 

522  528  534 

540 

546  552  5.58 

564  570  576 

582  5SS  594 

7 

560 

.567  574  581 

5SS  595  602 

609  616  623 

630 

637  644  651 

658  665  672 

679  686  593 

S 

640 

648  656  664 

672  680  6S8 

696  704  712 

720 

728  73f.  744 

752  760  768 

776  784  792 

9 

720 

729  738  747 

756  765  774 

783  792  801 

810 

819  82S  837 

846  855  864 

873  SS2  891 

14  DAY  USE 

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